Motivated by Lutwak et al.’s L_p-dual curvature measures, we introduce the concept of L_p-curvature measures. This new L_p-curvature measure is an extension of the classical surface area measure, L_p-surface area measure, and curvature measure. In this paper, we first prove some properties of the L_p-curvature measure. Next, using the L_p-curvature measure, we define the L_p,q-mixed volume which includes L_p-mixed volume as the special cases. Further, the Minkowski-type inequality related L_p,q-mixed volume and the uniqueness of the solution for the L_p,q- Minkowski problem are obtained. Finally, we propose several problems that need to be studied further.

1. Introduction

Surface area measure and integral curvature measure are two important measures in classical Brunn-Minkowski theory. Minkowski problem describing surface area measure and Aleksandrov problem describing integral curvature are two famous problems. As a generalization, L_p-surface area measure and L_p-integral curvature are defined in [1, 2], respectively. At the same time, the hyperbolic measure as the curvature measure of dual Fiedler is constructed in [3]. Lutwak et al. introduce L_p-dual curvature measure in [4], which is a generalization of the dual curvature, L_p-surface area measure and L_p-integral curvature. L_p-dual mixed volume (also known as (p, q)-dual mixed volume) is defined by [4] and Minkowski inequality is established. Furthermore, they study the L_p-dual Minkowski problem of L_p-dual curvature measure by reference to [5].

Inspired by Lutwak et al.’s L_p-dual curvature measure, a new concept of L_p-curvature measure is introduced in this paper. It includes classical surface area measure, L_p-surface area measure and curvature measure. In this paper, we first prove some properties of L_p-curvature measure. Next, based on L_p-curvature measure, we define L_p,q-mixed volume, which includes L_p-mixed volume as a special case. Furthermore, the Minkowski inequality for L_p,q-mixed volume and the uniqueness of the solution for L_p,q-Minkowski problem are obtained. Finally, some problems which need further study are put forward.

Let represent the set of convex bodies in n-dimensional Euclidean (compact convex subsets with nonempty embedding) space ℝⁿ, for convex bodies containing the origin inside in ℝⁿ, we write . Set B said centered on the origin of the unit sphere, B surface written as Sⁿ⁻¹, in ℝⁿ. V(K) represents the n dimensional volume of the body K and writes V(B) = ω_n.

For K ∈ Kⁿ, its support function, h_K = h(K, ·): ℝⁿ⟶(−∞, +∞), is defined by (see [6])

(1)

where x · y denotes the standard inner product of x and y.

For

and s, t ≥ 0 (not both zero), the Minkowski combination,

, of K and L is defined by the following:

(2)

i.e., sK + tL = {sx + ty : x ∈ K, y ∈ L}.

The surface area measure S(K, ·) of

can be defined by the following:

(3)

for any

. From Equation (3), the Minkowski’s first mixed volume of K and L is given as follows:

(4)

The mixed volume V₁(K, L) generalizes the concepts of volume, surface area, and mean width.

We say that

has a positive continuous curvature function f(K, ·) = f_K(·): Sⁿ⁻¹⟶ℝ, if for all

(5)

where S is spherical Lebesgue measure. Clearly, Equations (4) and (5) imply the following:

(6)

Let p ≥ 1. Using the L_p-Minkowski conbinations (see Equation (60)), Lutwak [2] defined the L_p-surface area measure S_p(K, ·) of a convex body

, namely, for each

(7)

For

, the L_p-mixed volume V_p(K, L) is given by the following (see [4]):

(8)

We say that

has a positive continuous L_p-curvature function f_p(K, ·): Sⁿ⁻¹⟶ℝ, if the integral representation

(9)

for all

. For

with a positive continuous curvature functions, it follows from Equation (8) and Equation (9) that

(10)

The L_p-Minkowski inequality of the L_p-mixed volume is (see [2, 7]) that for p ≥ 1,

(11)

with equality for p > 1 if and only if K and L are dilates, for p = 1 and if and only if K and L are homothetic.

According to Equation (10), the curvature function of L_p is the Radon-Nikodym derivative of L_p-surface area measure with respect to the spherical Lebesgue measure. The integral of L_p-curvature function (raised to an appropriate power) over the unit sphere is the L_p-affine surface area, which is an important research point of affine geometry and valuation theory, see, e.g., [8–24]. The L_p-Minkowski problem (see [2]) is a necessary and sufficient condition to find a given measure such that it is only the L_p-surface area measure of a convex body. Solving the L_p-Minkowski problem requires solving a degenerate singular Monge-Ampère-type equation on the unit sphere. The L_p-Minkowski problem has been solved for p ≥ 1, see [2, 25, 26], but critical cases for p < 1 remain open, see, e.g., [25, 27–31]. For its applications, see [5, 7, 27, 32–35].

A star body Q ⊂ ℝⁿ is a compact star-shaped set about the origin whose radial function ρ_Q : Sⁿ⁻¹⟶(0, ∞) is defined by the following:

(12)

for u ∈ Sⁿ⁻¹. If ρ(K, ·) is positive and continuous, K will be called a star body. Denote the set of star bodies in ℝⁿ by

. Obviously,

The dual Brunn-Minkowski theory is the theory of dual mixed volumes of star bodies. For q ∈ ℝ, the q-th dual mixed volume,

, of

is defined by the following:

(13)

where the integration is with respect to spherical Lebesgue measure. For q ≠ 0, the q-th dual volume

is defined by

. The q-th dual volume is important in geometric tomography, one of the reasons that is that for integers q = 1, 2, ⋯, n − 1 and each

(14)

where vol_q denotes volume in ℝ^q, G(n, q)(q = 1, 2, ⋯, n − 1) denote the Grassmann manifold of q-dimensional subspaces of ℝⁿ, the integration is with respect to the rotation invariant probability measure on G(n, q) and constant c_n,q is trivially determined by taking K to be B.

For the real q ≠ 0, the q-th dual curvature

is a Borel measure on the unit sphere, which can be defined in [3] by using the variational formula:

(15)

for every

. Similar to the critical role as L_p-surface area measures playing in the L_p Brunn-Minkowski theory, dual curvature measures is a central concept within the dual Brunn-Minkowski theory.

The singularity case q = 0 of dual volume leads to dual entropy of star body. For

, the dual entropy

can be defined as follows:

(16)

The L_p-integral curvature, J_p(K, ·), of

(see [1]) can be defined by a variational formula:

(17)

for all

, where K^∗ is the polar body of K is given by K^∗ = {x ∈ ℝⁿ : x · y ≤ 1 for all y ∈ K}.

In [4], Lutwak et al. introduced L_p-dual curvature measures, which are a generalization of dual curvatures, L_p-surface area measure and L_p-integral curvatures. For p, q ∈ ℝ,

and

, the L_p-dual curvature measure,

, is the Borel measure on Sⁿ⁻¹ defined by the following:

(18)

for each continuous g : Sⁿ⁻¹⟶ℝ, where α_K is the radial Gauss map (see Section 2 for details).

L_p-dual mixed volume (also known as (p, q)-dual mixed volume) is defined by Lutwak et al. [4] using the L_p-dual curvature:

For p, q ∈ ℝ,

, and

, the L_p-dual mixed volume

is defined by the following:

(19)

By Equation (18), the L_p-dual mixed volume has the following integral formula:

(20)

Specifically,

, namely,

(21)

and for

(22)

For the (p, q)-dual mixed volumes, the related Minkowski inequality is given in [4]. Suppose 1 ≤ q/n ≤ p, if

and

, then

(23)

with equality when q > n if and only if K, L and Q are dilates; while when q = n and p > 1, with equality if and only if K and L are dilates; while when q = n and p = 1, with equality if and only if K and L are homothetic.

In [4], the authors studied the L_p-dual Minkowski problems for L_p-dual curvature measures. The results of L_p-dual Minkowski problem caught many attentions, for example, see [3, 27, 36–42]. In addition, based on the (p, q)-dual mixed volumes, Ma et al. studied (p, q)-John ellipsoids in [43], which contain the classical John ellipsoid and the L_p-John ellipsoids. They also solved two involving optimization problem about the (p, q)-dual mixed volumes for all 0 < p ≤ q. A different extension of the L_p-John ellipsoid was considered by Li et al. in [44].

In this paper, motivated by Lutwak et al.’s works in [4], we introduce the following L_p-curvature measures which is a new curvature measure.

Definition 1. For p, q ∈ ℝ and , we define the L_p-curvature measure C_p,q(K, Q, ·) by the following:

(24)

for each continuous g : Sⁿ⁻¹⟶ℝ.

According to Definition 1, the L_p-curvature measure C_p,q(K, Q, ·) has the following integral expression.

Property 2. Suppose p, q ∈ ℝ. If , then

(25)

for each Borel set η⊆Sⁿ⁻¹. Here,

(26)

and H_K(v) is the supporting hyperplane to K with outer normal vector v ∈ ℝⁿ\{0}.

Property 3. Suppose p, q ∈ ℝ. If , then for each Borel set η⊆Sⁿ⁻¹,

(27)

Among them, represents the (n − 1)-dimensional Hausdorff measure, and ν_K(x) represents the regular radial vector of x ∈ ∂K, as well as x_K(η) represents the reverse spherical image of η ⊂ Sⁿ⁻¹.

The L_p-curvature measures unify the surface area measures, L_p-surface area measures and curvature measures, as well as other measures. In particular, for p, q ∈ ℝ and K,

the L_p-surface area measures and the q-th curvature measures (see Section 3 for its definition) are special cases of the L_p-curvature measures:

(28)

(29)

(30)

(31)

(32)

According to the L_p-curvature measures, we now define the notion of the L_p,q-mixed volumes which unifies L_p-mixed volumes and dual-mixed volumes.

Definition 4. For p, q ∈ ℝ and , the L_p,q-mixed volume, V_p,q(K, L, Q), of K and L (with respect to Q) is defined by the following:

(33)

The following variational formula is an extension of Equations (3) and (7).

Theorem 5. If reals p, q ≠ 0 and , then the L_p,q-mixed volume V_p,q(K, L, Q) via the variational formula of K and L (with respect to Q) by the following:

(34)

Using Equation (24), the L_p,q-mixed volume can be written by the following integral formula:

(35)

It will be shown that the L_p-mixed volume (Equation (8)) is the special case of the L_p,q-mixed volumes of convex bodies, i.e.,

(36)

The Minkowski-type inequality for L_p,q-mixed volume is as follows:

Theorem 6. Let and q ≥ 1, p < 0. Then,

(37)

with equality if and only if K, L, Q are dilates when q > 1 and K, Q are homothetic when q = 1.

For

, we say that the convex body Q with respect to K has a positive continuous (p, q)-curvature function f_p,q(K, Q, ·): Sⁿ⁻¹⟶ℝ, if

(38)

for all

. From Equations (33) and (38), we get that for

with a positive continuous curvature functions and a fixed

(39)

For q ∈ ℝ and t ∈ (0, ∞), the normalized power function

can be defined by the following:

(40)

For q ∈ ℝ and

, the normalized L_p-mixed volume

is defined by the following:

(41)

Note that for q ≠ 0, we have , while for p = 0 the normalized L_p-mixed volume is not just V_p(K, Q) multiplied by a constant but it can be considered from the mixed entropy (see Section 2 for details).

Another aim of this paper is to show that for p, q ∈ ℝ and

, there exists a variational formula that defines the L_p-curvature measure C_p,q(K, Q, ·) by the following:

(42)

for every

. This plays a key role to solve the associated Minkowski-type problems using a variational method.

Associated with L_p-curvature measures, (p, q)-Minkowski problem related to L_p-curvature measure asks: For a given Borel measure ϕ on a sphere, what are the necessary and sufficient conditions for the existence of a K convex body whose L_p-curvature measure is ϕ? The uniqueness of the problem is to ask to what extent is a convex body uniquely determined by its L_p-curvature measure?

The new (p, q)-Minkowski problem is equivalent to a degenerate singular Monge-Ampère equation on Sⁿ⁻¹: For fixed p, q ∈ ℝ,

(43)

where f : Sⁿ⁻¹⟶[0, ∞) is the given “data” function, h : Sⁿ⁻¹⟶(0, ∞) is the unknown function, and ℓ : Sⁿ⁻¹⟶Sⁿ⁻¹ is the identity map. Here,

and

denote the gradient vector and the Hessian matrix of h, respectively, with respect to an orthonormal frame on Sⁿ⁻¹, and I is the identity matrix. If we assume that the range of the gradient function

is D, then ν : D⟶Sⁿ⁻¹ is also an unknown function related to h.

Finally, we propose some problems that need further study, i.e., L_p,q-affine surface area problem, L_p,q-geominimal surface area problem and L_p,q-John ellipsoid problem.

2. Preliminaries

2.1. Basics in Convex Geometry

We work in the n-dimensional Euclidean space ℝⁿ. For x, y ∈ ℝⁿ, we use x · y to denote the standard inner product of x and y, and to denote the Euclidean norm of x. For x ∈ ℝⁿ\{0}, we will use both and 〈x〉 to abbreviate x/|x|.

We denote by C(Sⁿ⁻¹) the family of continuous functions defined on Sⁿ⁻¹ as endowed with the topology induced by the max-norm: , for f ∈ C(Sⁿ⁻¹).

For the support function, we know that for λ > 0 and x ∈ ℝⁿ,

(44)

Generally, for ϕ ∈ GL(n), the image ϕK = {ϕx : x ∈ K} satisfies that for x ∈ ℝⁿ,

(45)

where ϕ^t denotes the transpose of ϕ.

Since the support function is positive homogeneous of degree 1, we can restricted it on the unit sphere. For convex bodies

, their Hausdorff metric is given by the following:

(46)

At the point v ∈ Sⁿ⁻¹ where h_K is differentiable, the gradient of h_K in ℝⁿ is as follows:

(47)

where

denotes the gradient of h_K on Sⁿ⁻¹ with respect to the standard metric of Sⁿ⁻¹.

For the radial function, we see that for

, ϕ ∈ GL(n) and x ∈ ℝⁿ\{0},

(48)

Using the radial function, the volume of

can be expressed as follows:

(49)

For

, the polar body K^∗ of K is defined by the following:

(50)

From this definition, we get that for u ∈ Sⁿ⁻¹,

(51)

and for

(52)

For

, the Minkowski function of K is defined by the following:

(53)

Obviously, it is a continuous function on ℝⁿ, and

(54)

In the whole process, Ω ⊂ Sⁿ⁻¹ will represent a closed set that cannot be contained in any of the closed hemispheres of Sⁿ⁻¹. Wulff shape

, a continuous function h : Ω⟶(0, ∞), also known as h of the Aleksandrov body, is defined by the following:

(55)

, then it is easily seen that

(56)

Assume that the function ρ : Ω⟶(0, ∞) is continuous. Since Ω ⊂ Sⁿ⁻¹ is assumed to be closed, and ρ is continuous, we have {ρ(u)u : u ∈ Ω} is a compact set in ℝⁿ. The convex hull 〈ρ〉 generated by ρ,

(57)

is compact as well (see Schneider [40], Theorem 1.1.11). Since Ω is not contained in any closed hemisphere of Sⁿ⁻¹, we get that 〈ρ〉 contains the origin in its interior; namely,

. Obviously, if

(58)

The following lemma will be required.

Lemma 7 (see [3].)Let Ω ⊂ Sⁿ⁻¹ be a closed set that is not contained in any closed hemisphere of Sⁿ⁻¹. Let h : Ω⟶(0, ∞) be continuous. Then, the Wulff shape [h] determined by h and the convex hull 〈1/h〉 generated by the function 1/h are polar reciprocals of each other; namely,

(59)

Let and p ≥ 1. The L_p-Minkowski combination s · K+_pt · L is the convex body whose support function is given by the following (see [2]):

(60)

From Equation (53), we can extend the L_p-Minkowski combinations to the cases of p < 1.

Let p ≠ 0. For

, and s, t ∈ ℝ such that

is a strictly positive function on Sⁿ⁻¹, Lutwak et al. [4] defined the L_p-Minkowski combination

by the following:

(61)

When p = 0, define s · K+₀t · L by the following:

(62)

Note that s · K+₀t · L is defined for all s, t ∈ ℝ, since h_K, h_L are strictly positive functions on Sⁿ⁻¹.

Given ϕ ∈ SL(n) and p ≠ 0 (see [4]), we obtain that for s, t ∈ ℝ,

(63)

If s + t = 1, then Equation (63) holds for p = 0 as well.

For p ∈ ℝ\{0} and

, the L_p-mixed volume V_p(K, L) is defined by the following:

(64)

From Equations (64) and (63), we get that for ϕ ∈ SL(n) (see [45]),

(65)

The L_p-surface area S_p(K) of is given by S_p(K) = nV_p(K, B).

The following definition will be required.

Definition 8 (see [4].)Let p ∈ ℝ. If μ is a Borel measure on Sⁿ⁻¹ and ϕ ∈ SL(n), then ϕ_p⊣ μ, the L_p image of μ under ϕ, is a Borel measure such that

(66)

for each Borel f : Sⁿ⁻¹⟶ℝ.

Recall that the L_p-mixed volume has a dual integral formulation (see [4]): If

, then

(67)

where α_K is the radial Gauss map of K.

For

and p > 0, we define the volume-normalized L_p-mixed volume by the following:

(68)

Note that

is the normalized dual conical measure of K, it is a probability measure on suppS(·). Let p⟶0. Then,

(69)

The mixed entropy E(K, L) of

is defined by the following:

(70)

Note that

. As the case in Equation (63), for the dual mixed entropy, we have that for ϕ ∈ SL(n),

(71)

2.2. The Radial Gauss Map

The following results come from the articles [3, 4].

Suppose K is a convex body in ℝⁿ. For each v ∈ ℝⁿ\{0}, the hyperplane

(72)

is called the supporting hyperplane to K with outer normal vector v.

The spherical image of σ ⊂ ∂K is defined by the following:

(73)

The reverse spherical image of η ⊂ Sⁿ⁻¹ is defined by the following:

(74)

Suppose σ_K ⊂ ∂K is a set consisting of all x ∈ ∂K, for which the set ν_K({x}), which we frequently abbreviate as ν_K(x), contains more than a single element. It is a well-known fact that

(see Schneider [46], p. 84). The function on the set of regular radial vectors of ∂K is precisely defined by the following:

(75)

by making ν_K(x) be the unique element in ν_K(x) for each x ∈ ∂K\σ_K, The function ν_K is called the spherical image map of K and is known to be continuous (see Schneider [40], Lemma 2.2.12). It will be very convenient to abbreviate ∂K\σ_K by ∂^′K. Since

, when the integration is about

, it does not matter if the domain is over subsets of ∂^′K or ∂K.

The set η_K ⊂ Sⁿ⁻¹ consisting of all v ∈ Sⁿ⁻¹, for which the set x_K(v) contains more than a single element, is of

–measure 0 (see Schneider [40], Theorem 2.2.11). The function is precisely defined on the set of regular unit normal vectors of K:

(76)

by making x_K(v) be the unique element in x_K(v), for each v ∈ Sⁿ⁻¹\η_K. The function x_K is called the reverse spherical image map and is well known to be continuous (see Schneider [40], Lemma 2.2.12). By extending x_K to be a homogeneous function of degree 0 in ℝⁿ\{0}, we get a natural definition of x_K on the set of all regular normal vectors on ∂K.

For ω ⊂ Sⁿ⁻¹, the radial Gauss image of ω is defined by the following:

(77)

For a subset η ⊂ Sⁿ⁻¹, the reverse radial Gauss image of η is defined by the following:

(78)

Thus,

(79)

In particular, we can see that if η contains only a single vector v ∈ Sⁿ⁻¹,

(80)

Note that Equation (78), and hence for u ∈ Sⁿ⁻¹ and η ⊂ Sⁿ⁻¹, we see from Equation (77) that

(81)

Thus, for η₁, η₂⊆Sⁿ⁻¹,

(82)

We shall need to make use of the fact that for u, v ∈ Sⁿ⁻¹,

(83)

, then α_K(u) = {α_K(u)}, and Equation (77) becomes

(84)

and hence Equation (84) holds for almost all u ∈ Sⁿ⁻¹, with respect to spherical Lebesgue measure.

The following lemma will be used.

Lemma 9 (see [4].)If , then

(85)

for each η⊆Sⁿ⁻¹.

Since

for almost all v ∈ Sⁿ⁻¹ with respect to spherical Lebesgue measure, and

for almost all v ∈ Sⁿ⁻¹ with respect to spherical Lebesgue measure, Lemma 9 implies that if

, then

(86)

almost everywhere with respect to spherical Lebesgue measure.

For

, the radial map of K is defined by the following:

(87)

for u ∈ Sⁿ⁻¹. Note that

is just the restriction to ∂K of the map ℝⁿ\{0}⟶Sⁿ⁻¹.

The radial Gauss map of the convex body

is defined by the following:

(88)

where

. Since

is a bi-Lipschitz map between the spaces ∂K and Sⁿ⁻¹, so it follows that ω_K has spherical Lebesgue measure 0. We observed that if u ∈ Sⁿ⁻¹/ω_K, then α_K(u) contains only the element α_K(u). Since both ν_K and r_K are continuous, α_K is continuous. Notice that for x ∈ ∂^′K,

(89)

and hence for x ∈ ∂^′K,

(90)

If u ∈ Sⁿ⁻¹/ω_K, we see that x = ρ_K(u)u ∈ ∂K/ω_K with

from the definition of ω_K. Hence from Equation (89) we have

and we get the following (see [4]):

(91)

Combining with Equations (86) and (91), we have the following:

(92)

for almost all v with respect to spherical Lebesgue measure.

The surface area measure S(K, ·) of a convex body K can be defined, for Borel η⊆Sⁿ⁻¹, by the following:

(93)

where x_K(η) is the reverse spherical image of η ⊂ Sⁿ⁻¹.

If the boundary of a convex body K, denoted by ∂K, is smooth with positive Gauss curvature, the surface area measure of K is absolutely continuous with respect to spherical Lebesgue measure. The density can be regarded as the reciprocal of Gauss curvature and expressed in terms of the support function and its Hessian matrix on Sⁿ⁻¹:

(94)

where

denotes the Hessian matrix of h_K and I is the identity matrix with respect to an orthonormal frame on Sⁿ⁻¹. See Schneider [46].

For p ∈ ℝ and

, its L_p-surface area measure S_p(K, ·) introduced in [2] is defined by the following:

(95)

or equivalently by the following:

(96)

for each Borel η⊆Sⁿ⁻¹, where ν_K is the spherical image function of σ ⊂ ∂K.

For λ > 0, we easily see h_λK = λh_K and S(λK, ·) = λⁿ⁻¹S(K, ·). Then, Equation (91) implies the following:

(97)

The following integral identity is established in [3].

Lemma 10. If q ∈ ℝ and , while f : Sⁿ⁻¹⟶ℝ is bounded and Lebesgue integrable, then

(98)

In [3], we see that

Lemma 11. If is strictly convex, and f : Sⁿ⁻¹⟶ℝ and F : ∂K⟶ℝ are both continuous, then

(99)

where ∇h_K is the gradient of h_K in ℝⁿ, and ν_K is defined only on ∂K\σ_K, the set σ_K has

measure 0.

We will require a slight extension of Equation (97). To be specific, if p ∈ ℝ, while

is strictly convex, and f : Sⁿ⁻¹⟶ℝ and F : ∂K⟶ℝ are both continuous, then (see [4])

(100)

The following lemma will be used.

Lemma 12 (see [4].)For each p ∈ ℝ, the set

(101)

is dense in C(Sⁿ⁻¹).

3. L_p-Curvature Measures

For a star body

, define ‖·‖_Q : ℝⁿ⟶[0, ∞) by letting (see [4])

(102)

Note that ‖·‖_Q is continuous and positively homogeneous of degree 1. If Q is an origin-symmetric convex body in ℝⁿ, then ‖·‖_Q is just an ordinary norm in ℝⁿ, and (ℝⁿ, ‖·‖_Q) is the n-dimensional Banach space whose unit ball is Q.

Note that the definition (Equation (102)) is an extension of Minkowski functional (Equation (53)) of convex body .

Definition 13. Suppose q ∈ ℝ. For , the q-th area measure S_q(K, Q, ·) is defined by the following:

(103)

for each Lebesgue measurable ω⊆Sⁿ⁻¹, and the q-th curvature measure C_q(K, Q, ·) is defined by the following:

(104)

for each Borel η⊆Sⁿ⁻¹. Moreover, for each p ∈ ℝ, the L_p-curvature measure C_p,q(K, Q, ·) is defined by the following:

(105)

Observe that

(106)

Note that from definition (Equation (104)) and the fact that Equation (84) holds off of the set ω_K of spherical Lebesgue measure 0, so for each Borel η⊆Sⁿ⁻¹, we get the following:

(107)

That is,

(108)

We observed that C_q(K, Q, ·) is absolutely continuous with respect to spherical Lebesgue measure. Then, from Equation (108), we deduce that

Lemma 14. Let and q ∈ ℝ. If each function f : Sⁿ⁻¹⟶ℝ is bounded and Borel, then

(109)

Proof. Because Equation (109) is shown by Equation (108) as an indicator function of the Borel set, we see that Equation (109) holds for a linear combination of the indicator functions of the Borel set, namely, simple functions ϕ : Sⁿ⁻¹⟶ℝ, is given by the following:

(110)

where c_i ∈ ℝ and Borel η_i ⊂ Sⁿ⁻¹. Now let us choose a sequence of simple functions ϕ_k : Sⁿ⁻¹⟶ℝ converging to the bounded Borel function f : Sⁿ⁻¹⟶ℝ. Note that f is bounded, ϕ_k can be selected as uniformly bounded. Then, ϕ_k∘α_K converges pointwise to f∘α_K on Sⁿ⁻¹\ω_K. Since f : Sⁿ⁻¹⟶ℝ is a Borel function and the radial Gauss map α_K : Sⁿ⁻¹\ω_K⟶Sⁿ⁻¹ is continuous; thus, f∘α_K is a Borel function on Sⁿ⁻¹\ω_K. Because f is bounded, and ω_K has spherical Lebesgue measure 0, we can infer that f is C_q(K, q, ·) integrable, and f∘α_K is spherical Lebesgue integrable in Sⁿ⁻¹. Since C_q(K, q, ·) is a finite measure, by taking the limit k⟶∞, we obtain Equation (109).☐

Proposition 15. Let p, q ∈ ℝ. If , then

(111)

for each Borel set η⊆Sⁿ⁻¹.

Proof. From Equations (105), (109), and (84), we have for each Borel η⊆Sⁿ⁻¹,

(112)

☐

Obviously, the total measures of the q-th curvature measure and the q-th area measure are the q-th mixed volume, i.e.,

(113)

It follows immediately from Equations (103) and (104) that

(114)

The L_p-curvature measures have the following properties.

Property 16. Let p, q ∈ ℝ. If . Then, for each Borel set η⊆Sⁿ⁻¹ and each bounded Borel function g : Sⁿ⁻¹⟶ℝ, we have the following:

(115)

(116)

(117)

Proof. Because is a bounded Borel function, from Equation (109) with , we have the following:

(118)

Thus, in light of Equation (105) is the desired result (Equation (115)).

By Equations (115), (89), and (90), and letting and q = n in Equation (98), we have the following:

(119)

☐

This yields Equation (116).

Take g = 1_η in Equation (116). Notice that ν_K(x) ∈ η⇔x ∈ x_K(η) for almost all x with respect to spherical Lebesgue measure. So, we immediately obtain Equation (117).

Remark 17. Equation (115) tells us the rationality for Definition 1 of the L_p-curvature measure C_p,q(K, Q, ·).

Example 18 (Lp-curvature measures of polytopes). Suppose be a polytope with outer unit normal vectors v₁, v₂, ⋯, v_m. If Δ_i is a cone consisting of all rays emanating from the origin and passing through the face of P whose outer normal is v_i. Remember that we abbreviate by , and from Equation (80), we get the following:

(120)

If η ⊂ Sⁿ⁻¹ is a Borel set such that {v₁, v₂, ⋯, v_m}∩η = ∅, then has spherical Lebesgue measure 0. So, the L_p-curvature measure C_p,q(P, Q, ·) is discrete and concentrated on {v₁, v₂, ⋯, v_m}. From Proposition 15 and Equation (120), we have the following:

(121)

where

represents the delta measure centered on v_i, and

(122)

Example 19 (Lp-curvature measures of strictly convex bodies). Let are strictly convex. Suppose g : Sⁿ⁻¹⟶ℝ is continuous, then we start with Equations (116) and (100)(taking ) and combine the fact that ∂K/∂^′K has measure 0, it follows that

(123)

Using Equation (95), this shows that

(124)

Example 20 (Lp-curvature measures of smooth convex bodies). Let has a C² boundary with everywhere positive Gauss curvature. Because in this case, S(Q, ·) is absolutely continuous for the spherical Lebesgue measure; therefore, C_p,q(K, Q, ·) is absolutely continuous for the spherical Lebesgue measure, and from Equations (124), (94), and (47), we get the following:

(125)

where

represents the gradient of h_K on Sⁿ⁻¹ at v and

represents the Hessian matrix of h_K with respect to an orthonormal frame on Sⁿ⁻¹. We write Equation (125) as 1/nf_p,q(K, Q, v), that is,

(126)

We say convex body Q with respect to a fixed convex body K as a parameter have a positive continuous (p, q)-curvature function f_p,q(K, Q, ·).

The weak convergence of L_p-curvature measure is an important property contained in the following propositions.

Proposition 21. Let p, q ∈ ℝ and . If with , then C_p,q(K_i, Q, ·)⟶C_p,q(K₀, Q, ·), weakly.

Proof. Let g : Sⁿ⁻¹⟶ℝ is continuous. From Equation (115) we know that

(127)

for all i. Since K_i⟶K₀, with respect to the Hausdorff metric, we have that

, uniformly on Sⁿ⁻¹, and the surface area measure has the following property (see [2, 7, 23]):

(128)

Thus,

(129)

It follows that C_p,q(K_i, Q, ·)⟶C_p,q(K₀, Q, ·), weakly. ☐

The following statement contains the absolute continuity of L_p-curvature measure with respect to surface area measure.

Proposition 22. Let p, q ∈ ℝ. If , then L_p-curvature measure C_p,q(K, Q, ·) is absolutely continuous with respect to the surface area measure S(K, ·).

Proof. Let η ⊂ Sⁿ⁻¹ be such that S(K, η) = 0, or equivalently by definition (Equation (96)), . Then, Equation (117) states that

(130)

Thus, the integration is over a set of measure 0.☐

The following proposition shows that the L_p-curvature measure including the classical surface area measures and the L_p-surface area measures. Therefore, the classical surface area measures and the L_p-surface area measures are special cases of the L_p-curvature measures.

Proposition 23. Suppose and p, q ∈ ℝ. Then,

(131)

(132)

(133)

(134)

(135)

Proof. Let η ⊂ Sⁿ⁻¹ be a Borel set. From Equations (117) and (96), we have the following:

(136)

Therefore, we get Equations (131) and (133).

From Equations (117), (54), (90), and (96), we have the following:

(137)

Therefore, we get Equation (132). Similarly, we can get the rest.☐

Recall that the concept of the valuation. A function Φ defined on the space

of convex bodies and taking values in an abelian semigroup is called a valuation if

(138)

whenever

The set of Borel measures on Sⁿ⁻¹ is represented by

. We are going to prove that now, for fixed indices p, q ∈ ℝ, and a fixed convex body

, the functional

, defined by Q ↦ C_p,q(K, Q, ·) is a valuation; namely, if

, are such that

then

(139)

To prove the valuation of L_p-curvature measure, we shall employ Weil’s approximation lemma (see [4]):

Lemma 24. If are such that K ∪ L is convex, then K and L may be approximated by sequences of bodies that are both strictly convex and smooth and such that .

We appeal to Proposition 21 together with Weil’s approximation lemma in order to complete our proof.

Theorem 25. Suppose p, q ∈ ℝ and . Then, the functional

(140)

defined by K ↦ C_p,q(K, Q, ·), is a valuation.

Proof. We will use the fact that if are such that , then h_K∪L = max{h_K, h_L} and h_K∩L = min{h_K, h_L}. We will also take advantage of the fact that ν_K and ν_L are defined almost everywhere on the boundaries of K and L, respectively.

First of all, let us assume that K and L are both strictly convex. For a fixed θ ⊂ Sⁿ⁻¹, write θ as the union of three disjoint pieces θ = θ₀ ∪ θ_K ∪ θ_L, where

(141)

while

(142)

In this case, we have the following:

(143)

while

(144)

Alternatively, using Equation (117), this has

(145)

Similarly,

(146)

It is also the case that

(147)

In order to see the fact that the last one, we observe that the strict convexity of K and L forces x_K∪L(θ₀) = x_K∩L(θ₀).

Using the fact that C_p,q(K, ·, ·) is a measure in the third argument on Sⁿ⁻¹, combined with the fact that the union θ = θ₀ ∪ θ_K ∪ θ_L is disjoint, by adding Equations (145), (146), and (147) we obtain that

(148)

which is the desired result.

For any , we resort to Proposition 21 in order to use the weak continuity of C_p,q(·, Q, ·) in the first argument.☐

4. Variational Formulas for L_p,q-Mixed Volumes

Suppose Ω is a closed subset of Sⁿ⁻¹ that is not contained in any closed hemisphere. Let h₀ : Ω⟶(0, ∞) and f : Ω⟶ℝ be consecutive, and δ > 0. Let h_t : Ω⟶(0, ∞) be a positive continuous function defined as follows:

(149)

for each t ∈ (−δ, δ), where o(t, ·): Ω⟶ℝ is continuous and

, uniformly on Ω. And denote by

(150)

Wulff shape determined by h_t. We call [h_t] the logarithmic Wulff shape family generated by (h₀, f). If h₀ is the support function h_K of convex body K, we also put [h_t] written [K, f, t].

Let ρ₀ : Ω⟶(0, ∞) and g : Ω⟶ℝ be continuous, and δ > 0. Let ρ_t : Ω⟶(0, ∞) be a positive continuous function defined by the following:

(151)

for each t ∈ (−δ, δ), where again o(t, ·): Ω⟶ℝ is continuous and

uniformly on Ω. And denote by

(152)

the convex hull generated by ρ_t. We call 〈ρ_t〉 the logarithmic family of convex hull generated by (ρ₀, g). If ρ₀ is the radial function ρ_K of convex body K, we also put 〈ρ_t〉 as 〈K, g, t〉.

The following lemma shows that the support functions of a logarithmic family of the polar of convex hulls are differentiable with respect to the variational variable.

Lemma 26. Suppose Ω ⊂ Sⁿ⁻¹ be a closed set that is not contained in any closed hemisphere of Sⁿ⁻¹. Let ρ₀ : Ω⟶(0, ∞) and g : Ω⟶ℝ be continuous. If 〈ρ_t〉 is a logarithmic family of convex hulls of (ρ₀, g) and q ∈ ℝ, then

(153)

for all

; namely, for all regular normals v of

, where Equation (153) holds a.e. with respect to spherical Lebesgue measure. Moreover, there exist δ > 0 and M > 0 so that

(154)

for all v ∈ Sⁿ⁻¹ and all t ∈ (−δ, δ).

Proof. Obviously,

(155)

Therefore,

(156)

Since 〈ρ₀〉 and are two convex bodies in , and as t⟶0, there exist m₀, m₁ ∈ (0, ∞) and δ₀ > 0 such that

(157)

for each t ∈ (−δ₀, δ₀). From this, it follows that there exists M₁ > 1 so that

(158)

It is easily seen that s − 1 ≥ logs whenever s ∈ (0, 1), whereas s − 1 ≤ M₁logs whenever s ∈ [1, M₁]. Thus,

(159)

It follows that

(160)

that is

(161)

on Sⁿ⁻¹, whenever t ∈ (−δ₀, δ₀).

Let . Since o(t, ·)/t⟶0 as t⟶0 uniformly on Ω, we may choose δ₁ > 0 so that for all t ∈ (−δ₁, δ₁), we have |o(t, ·)| ≤ |t| on Ω. From Equation (151) and the definition of M₀, we immediately see that

(162)

on Sⁿ⁻¹, whenever t ∈ (−δ₁, δ₁). Let (−δ, δ) = (−δ₀, δ₀)∩(−δ₁, δ₁). Together with Equations (161) and (162), we give Equation (154).☐

The following theorem gives variational formulas for the L_p-mixed volume and L_p-mixed entropy for a family of logarithmic convex hulls.

Theorem 27. Let Ω ⊂ Sⁿ⁻¹ is a closed set not contained in any closed hemisphere of Sⁿ⁻¹. If ρ₀ : Ω⟶(0, ∞) and g : Ω⟶ℝ are continuous, and 〈ρ_t〉 is a logarithmic family of convex hulls of (ρ₀, g), then for and q ≠ 0,

(163)

for q = 0,

(164)

Proof. Abbreviate by η₀. Recall that η₀ is the set of spherical Lebesgue measure zero that consists of the complement, in Sⁿ⁻¹, of the regular normal vectors of the convex body K^∗. Note that the continuous function

(165)

is well defined by

for all v ∈ Sⁿ⁻¹\η₀.

Let v ∈ Sⁿ⁻¹/η₀. To see that , let

(166)

for some u₀ ∈ Ω. This means that

(167)

and hence . Because in addition to obviously belonging to K^∗, it also belongs to . But v is a regular normal vector of K^∗, and therefore, . Then,

(168)

From this, Equation (168), Equation (52), and Lemma 9 yield the following facts:

(169)

As Ω is closed, by using the Tietze extension theorem, extend the continuous function g : Ω⟶ℝ to a continuous function . Therefore, using Equation (169) we see that

(170)

for v ∈ Sⁿ⁻¹\η₀.

Using Equation (22), the fact that η₀ has measure zero, Equation (51), Equation (154), the dominated convergence theorem, Lemma 26, Equation (86), Equation (170), Lemma 14, and again Equation (170), we have the following:

(171)

☐

According to Equations (70) and (51), the fact that η₀ has measure zero, the dominated convergence theorem, Equation (151), together with Equations (170) and (86), Lemma 14, and again Equation (170), we have the following:

(172)

Using the same argument as in the second part of the proof, we get that

(173)

The following theorem gives the variational formulas for the L_p-mixed volumes and mixed entropy of the logarithmic family of Wulff shapes.

Theorem 28. Suppose Ω ⊂ Sⁿ⁻¹ is a closed set not contained in any closed hemisphere of Sⁿ⁻¹. Let h₀ : Ω⟶(0, ∞) and f : Ω⟶ℝ be continuous, and [h_t] be a logarithmic family of Wulff shapes associated with (h₀, f). If , then for q ≠ 0,

(174)

for q = 0,

(175)

Proof. The logarithmic family of Wulff shape [h_t] is defined as the Wulff shape of h_t, where h_t is given by the following:

(176)

Let . Then,

(177)

Let 〈ρ_t〉 be the logarithmic family of convex hulls associated with (ρ₀, −f). Then from Lemma 7, we obtain that

(178)

and the desired conclusions now follow from Theorem 27.

We describe the special cases of Theorem 27 and Theorem 28 for logarithmic families of convex hull and Wulff shape generated by convex bodies.

Theorem 29. If and g : Sⁿ⁻¹⟶ℝ is continuous, then for q ≠ 0,

(179)

for q = 0,

(180)

Proof. In Theorem 27, let . Then, . In particular, from (53) we have .☐

Above variational formulas for convex hulls imply variational formulas for Wulff shapes.

Theorem 30. If and f : Sⁿ⁻¹⟶ℝ is continuous, then for q ≠ 0,

(181)

for q = 0,

(182)

Proof. The logarithmic family of Wulff shapes [Q, f, o, t] is defined by the Wulff shape [h_t], where

(183)

This, and the fact that , allows us to define

(184)

and

will generate a logarithmic family of convex hulls 〈Q^∗, −f, −o, t〉. Since

, Lemma 7 gives the following:

(185)

Therefore, Theorem 30 now follows directly from Theorem 29.☐

The following theorem gives the variational formulas of L_p-mixed volumes and mixed entropies with respect to L_p Minkowski combinations.

Theorem 31. If p, q ∈ ℝⁿ and , then for p ≠ 0, q ≠ 0,

(186)

for p = 0 and q ≠ 0,

(187)

for p ≠ 0 and q = 0,

(188)

(189)

and if p = q = 0,

(190)

Proof. For small t, h_t is defined by the following:

(191)

From Equations (61) and (62), the Wulff shape [h_t] = Q+_pt · L. For sufficiently small t, it follows from Equation (191) that

(192)

Let when p ≠ 0, and let f = logh_L when p = 0. The required formulas now follow Theorem 30 and Equation (105).☐

We use the normalized power function, and we can write the formula in Theorem 31 as a single formula.

Theorem 32. Suppose p, q ∈ ℝ. For ,

(193)

For L₀ Minkowski linear combinations, it would help to have an affine version of Theorem 31. This is contained in

Theorem 33. Suppose q ≠ 0. If , then

(194)

(195)

(196)

Proof. Let

(197)

From Equation (58) we know the Wulff space [h_t] = (1 − t) · Q+₀t · L. From the above definition of h_t, it follows immediately that for sufficiently small t,

(198)

Let f = logh_L/h_Q. The desired formulas now follow directly from Theorem 30.☐

Theorem 34. If p ≠ 0 and q ≠ 0, then for all and ϕ ∈ SL(n),

(199)

(200)

(201)

(202)

Proof. Obviously, the case p ≠ 0 and q = 0 is handled by Equation (200). The case p = 0 and q ≠ 0 is handled by Equation (201), while the case p = 0 and q = 0 is handled by Equation (202).

We adopt the methods and techniques of paper [4]. Recall that Haberl and Parapatits refer to the [9] classified measure-valued operators on , which are SL(n)-inverse degree p and corresponding to the transformation behavior in Theorem 34. From Equations (63), (65), and (186), we see that for all and all ϕ ∈ SL(n),

(203)

or equivalently for all

and all ϕ ∈ SL(n),

(204)

By Definition 8, and note the important fact that support functions are positively homogeneous of degree 1, from Equations (45) and (204), we have the following:

(205)

This shows that the measures and C_p,q(ϕK, ϕQ, ·) when integrated against the p-th power of support functions of bodies in are identical; thus, Lemma 12 now indicates that

(206)

it can be concluded that Equation (199).

The proof for Equation (200) is the same as the proof for Equation (199): As long as p ≠ 0, it will be the case that Equation (204) continues to hold even if q = 0 provided we appeal to Equations (188) and (71) when previously we had turned to Equations (188) and (65).

From Equations (63), (65), and (194), we know that for all and all ϕ ∈ SL(n),

(207)

In Equation (207), choose L = B. Then, by Equation (45), we see that , and (6.15) becomes the following form:

(208)

for all ϕ ∈ SL(n) and all

. Together with Equations (207) and (208), we have the following:

(209)

this and Equation (45) give that for all ϕ ∈ SL(n) and all

(210)

Equivalently,

(211)

for all ϕ ∈ SL(n) and all

. Using Lemma 12, we see that Equation (211) yields

(212)

for all ϕ ∈ SL(n) and all

. This establishes Equation (201).

The proof of Equation (202) is identical to the proof of Equation (201) except that instead of appealing to Equations (194) and (65) we appeal to Equations (195) and (71).☐

5. The L_p,q-Mixed Volumes

For

, the L_p-mixed volume V_p(K, L) has the integral representation

(213)

From Equation (115), with q = p and

, we have that

(214)

By Equation (131), the L_p-mixed volume V_p(K, L) has a dual integral formulation. If

, then

(215)

The dual integral formulation of L_p-mixed volume was first introduced by Lutwak et al. in [4]. This leads us to define following L_p,q-mixed volumes.

Definition 35. Let p, q ∈ ℝ and . The L_p,q-mixed volume V_p,q(K, L, Q) is defined by the following:

(216)

Using Equation (115) with

, Equation (216) can be written as follows:

(217)

From Equations (216) and (124), the L_p-mixed volume V_p,q(K, L, Q) can be written as follows:

(218)

where the function ν_K : {∇h_K(v): v ∈ Sⁿ⁻¹} ⊂ ∂^′K⟶Sⁿ⁻¹.

From L_p,q-mixed volume (Equation (30)) (or Equation (217)), the L_p-mixed volume (Equation (9)) (or Equation (22)) will be shown to be the special cases.

Proposition 36. Suppose p, q ∈ ℝ. If , then

(219)

(220)

(221)

(222)

(223)

Proof. Identity (Equations (219)–(221)) follow from Equation (22) and Equation (34) (or Equation (217)). Similarly, we can prove Equations (222) and (223).

Proposition 37. The L_p,q -mixed volume V_p,q is SL(n)-invariant. That is, for p, q ∈ ℝ, , and ϕ ∈ SL(n),

(224)

Proof. For p = 0, the conclusion follows from Equation (223) and the SL(n)-invariance of L_p-mixed volumes (Equation (65)). We assume p ≠ 0. By Definition 35, Equation (199), and Equation (200), the fact that support functions are positively homogeneous of degree 1, Equation (45), and Definition 8, we have the following:

(225)

From the dual Equation (217) of L_p,q-mixed volume and Equation (44), we have for real λ > 0,

(226)

Proposition 37, together with Equations (216) and (226), shows that for ϕ ∈ GL(n),

(227)

For L_p,q-mixed volume, the following inequality is a generalization of the L_p-Minkowski inequality for L_p-mixed volume.

Theorem 38. Suppose p, q are such that q ≥ 1 and p < 0. If , then

(228)

with equality if and only if K, L, Q are dilates when q > 1 and K, Q are homothetic when q = 1.

Proof. From Equations (21) and (217), we have the following:

(229)

From this, by the Hölder inequality (see [47]), the dual integral formulation (Equation (22)) of L_p-mixed volume and L_p-Minkowski inequality (Equation (11)), we have the following:

(230)

The equality conditions follow from the equality conditions of Hölder inequality and the L_p-Minkowski inequality (Equation (11)) for L_p-mixed volumes. Namely, the equality for the above inequality holds if and only if K, L, Q are dilates when q > 1 and K, Q are homothetic when q = 1.

Over the past three decades, valuation theory has become an ever more important part of convex body geometry. See, e.g., [11–13, 18, 48–53]. The convex L_p,q-mixed volume is the valuation for each entry.

Proposition 39. The L_p,q-mixed volume V_p,q(K, L, Q) is a valuation over with respect to all K, L, and Q.

Proof. The L_p,q-mixed volume V_p,q(K, L, Q) is a valuation on respect to the third argument can be seen easily by writing Equation (216) as follows:

(231)

and from Equation (139) (or Theorem 25), observing that for

, we have the following:

(232)

Together with Equations (216) and (232), we have the following:

(233)

Namely, V_p,q(K, L, Q) is a valuation in the third argument.

Observing that for such that . Then, we have the following:

(234)

Note that and . Together with Equations (216) and (234), we see that V_p,q(K, L, Q) is a valuation in the second argument, i.e,

(235)

Note that if are such that , then we have the following:

(236)

Together with Equations (218) and (236), we see that V_p,q(K, L, Q) is a valuation in the first argument, i.e,

(237)

Let . The q-th mixed cone-volume measure C_q(K, Q, ω) of K and Q is a Borel measure on the unit sphere Sⁿ⁻¹ is defined by for a Borel ω⊆Sⁿ⁻¹ and u ∈ ω,

(238)

Since the q-th mixed volume, V_q(K, Q) has a dual integral formulation:

(239)

We can turn the q-th mixed cone-volume measure into the probability measure on the unit sphere by normalizing it by q-th mixed volume of the bodies. The q-th mixed cone-volume probability measure of K and Q is defined by the following:

(240)

If , then for each real p, q ∈ ℝ, we define the normalized L_p,q-mixed volume by the following:

(241)

Let p⟶0. We give the following:

(242)

The q-th mixed entropy E_q(K, L, Q) of convex bodies is defined by the following:

(243)

In particular,

(244)

☐

6. The L_p,q-Minkowski Problems

The existence and uniqueness of L_p,q-Minkowski problem is the central problem to be investigated here. Its existence problem can be expressed as follows:

Problem 40. Let p, q ∈ ℝ, and is fixed. Given a Borel measure , what are necessary and sufficient conditions on μ such that there exists a whose L_p-curvature measures C_p,q(K, Q, ·) is the given measure μ?

L_p-Minkowski problem when q = p. When the given data measure μ has a density f, it follows from Equation (125) that L_p,q-Minkowski problem is equivalent to solving the following Monge-Ampère-type equation on Sⁿ⁻¹:

(245)

where h is the unknown function on Sⁿ⁻¹, and ∇h is the gradient vector function in ℝⁿ of the extension from h to ℝⁿ as a vector function that is positively homogeneous of degree 1. If we assume that the range of the gradient function ∇h is D, then ν : D⟶Sⁿ⁻¹ is also an unknown function related to h.

Our uniqueness result for the L_p,q-Minkowski problem is presented in the following:

Problem 41. For fixed p, q ∈ ℝ and , if such that

(246)

then how is K related to L?

Now, we establish uniqueness of the solution to the problem with q ≥ n for the case of polytopes.

Theorem 42. Let be polytopes and let . Suppose

(247)

Then, P = P^′ when q > n and P^′ is a dilate of P when q = n.

Proof. According to Equations (121) and (122), we get that the curvature measures of polytopes are discrete, and that C_p,q(P, Q, ·) = C_p,q(P^′, Q, ·) implies that P and P^′ must have the same outer unit normal vectors v₁, v₂, ⋯, v_m and

(248)

where

denotes the delta measure concentrated at v_i, and

(249)

Here Δ_i and Δ′_i are the cones formed by the origin and the facets of P and P^′ with vector v_i, respectively.

Assume that P ≠ P^′. Tt is easy to see that P⊆P^′ is not possible. Set λ be the maximal number with λP⊆P^′. This has λ < 1. Since λP and P^′ have the same outer unit normal vectors, there is a facet of λP which is contained in a facet of P^′. The outer unit normal vector of those facets is denoted by . It follows that

(250)

Thus,

(251)

with equality if and only if

. By this and Equation (249), we can obtain that

(252)

But λ < 1 implies that λ^n−q > 1 if q > n. Obviously, this is a contradiction.☐

If q = n, then Equation (249) forces equality in Equation (251). So, , and the facets of λP and P^′ with outer unit normal vector are the same. Let is the outer unit normal vector to a facet, which is adjacent to the facet whose outer unit normal vector is . Thus, the facet of λP with outer unit normal vector is contained in the facet of P^′ with outer unit normal vector . A similar argument holds that the two facets are the same. Continuing in this manner, it follows that λP = P^′.

7. Several Other Problems

Here, we present several issues that need to be discussed in the future. Some of the definitions and problems below are different from the paper [40, 43, 44, 54].

7.1. L_p,q-Mixed Affine Surface Areas

In [7], Lutwak defined the L_p-affine surface area Ω_p(K) for p ≥ 1 by the following:

(253)

Hug in [55] observed that the L_p-affine surface area is well defined for 0 < p < 1.

The following affine isoperimetric inequality was established in [7] for p ≥ 1, and in [56] for 0 < p < 1. If

, then

(254)

with equality if and only if K is an ellipsoid. Here, ω_n is the volume of the n dimensional unit sphere.

Definition 43. Suppose q ∈ ℝ. For and , the q-th curvature measure C_q(K, Q^∗, ·) of K (related to star body Q) is defined by the following:

(255)

for each Borel η⊆Sⁿ⁻¹, and L_p-curvature measure C_p,q(K, Q^∗, ·) of K is defined by the following:

(256)

for each Borel η⊆Sⁿ⁻¹.

It follows from Definition 43 that

(257)

Definition 44. Suppose q ∈ ℝ. If , the q-th mixed volume V_q(K, L^∗) is defined by the following:

(258)

Definition 45. Suppose p, q ∈ ℝ. If and , the L_p,q-mixed volume V_p,q(K, L^∗, Q^∗) of K and L^∗ (with respect to Q) is defined by the following:

(259)

Inspired by [40, 54], from Equations (258) and (259) we define L_p,q-mixed affine surface area as follows:

Definition 46. For p ∈ ℝ, q > 0 and , the L_p,q -mixed affine surface area Ω_p,q(K, Q) of K (relate to Q) is defined by the following:

(260)

When Q = L, from Equation (219) we have the following:

(261)

Ω_p,q(K, L) is the L_q-affine surface area Ω_q(K).

Problem 47. For the L_p,q-mixed affine surface area, does it maintain affine invariance and continuity? How to establish its affine isoperimetric inequality?

7.2. L_p,q-Mixed Geominimal Surface Area

In [7], Lutwak defined the L_p-geominimal surface area G_p(K) by the following:

(262)

and proved the following affine isoperimetric inequality: If

, then

(263)

with equality if and only if K is an ellipsoid.

Motivated by the L_p-mixed geominimal surface area (Equation (257)), we define L_p,q-mixed geominimal surface area, G_p,q(K, Q), of K relative to Q as follows:

Definition 48. For p ∈ ℝ, q ≥ 1, and , the L_p,q-mixed geominimal surface area G_p,q(K, Q) of K relative to Q is defined by the following:

(264)

When Q = L, from Equation (219) we have the following:

(265)

G_p,q(K, L) is the L_q-geominimal surface area G_q(K).

Problem 49. For the L_p,q-mixed geominimal surface area, does it maintain affine invariance and continuity? How to establish its affine isoperimetric inequality?

7.3. L_p,q-John Ellipsoids

Suppose p ∈ (0, ∞) and K is a convex body in ℝⁿ with the origin in its interior. Among all origin-symmetric ellipsoids E, the unique ellipsoid that solves the constrained maximization problem:

(266)

is called the L_p-John ellipsoid of K which defined in [45] and denoted by E_pK. Clearly, E_pB = B. Here,

(267)

is the normalized L_p-mixed volume of K and E. In the case p = ∞, we define the following:

(268)

In general, the L_p-John ellipsoid E_pK is not contained in K (except when p = ∞). However, when 1 ≤ p ≤ ∞, it has V(E_pK) ≤ V(K). In reverse, for 0 < p ≤ ∞, the L_p version of ball’s volume-ratio inequality [45] states that

(269)

with equality if and only if K is a parallelotope.

We know that from Equation (241), for 0 < p < ∞, q ∈ ℝ, the normalized L_p,q-mixed volume is calculated by the following:

(270)

In the case p = ∞, define the following:

(271)

By Equation (271), we have the following:

(272)

Let denote the class of origin-symmetric ellipsoids in ℝⁿ. Inspired by the constrained maximization problem (Equation (266)), the reader may consider its L_p,q-version.

Problem 50. Let 0 < p ≤ ∞, q ∈ ℝ. For , find an ellipsoid, among all origin-symmetric ellipsoids, which solves the following constrained maximization problem:

(273)

An ellipsoid that solves the constrained maximization problem will be called L_p,q-John ellipsoid for K, Q and denoted by E_p,q(K, Q).

In particular, when Q = K, from Equations (219) and (22), we have the following:

(274)

Thus, . So, Problem 50 degenerates into the problem.

Conflicts of Interest

The author declares that there are no competing interests.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11561020). The author is particularly grateful to Professor Weidong Wang, Dr. Yibin Feng, and Dr. Denghui Wu for their comments on various drafts of this work.

Open Research

Data Availability

All data included in this study are available upon request by contact with the corresponding author.

References

All articles

L_p-Curvature Measures and L_p,q-Mixed Volumes

Abstract

1. Introduction

2. Preliminaries

2.1. Basics in Convex Geometry

2.2. The Radial Gauss Map

3. L_p-Curvature Measures

4. Variational Formulas for L_p,q-Mixed Volumes

5. The L_p,q-Mixed Volumes

6. The L_p,q-Minkowski Problems

7. Several Other Problems

7.1. L_p,q-Mixed Affine Surface Areas

7.2. L_p,q-Mixed Geominimal Surface Area

7.3. L_p,q-John Ellipsoids

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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Lp-Curvature Measures and Lp,q-Mixed Volumes

Abstract

1. Introduction

2. Preliminaries

2.1. Basics in Convex Geometry

2.2. The Radial Gauss Map

3. Lp-Curvature Measures

4. Variational Formulas for Lp,q-Mixed Volumes

5. The Lp,q-Mixed Volumes

6. The Lp,q-Minkowski Problems

7. Several Other Problems

7.1. Lp,q-Mixed Affine Surface Areas

7.2. Lp,q-Mixed Geominimal Surface Area

7.3. Lp,q-John Ellipsoids

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

Related

Information

L_p-Curvature Measures and L_p,q-Mixed Volumes

3. L_p-Curvature Measures

4. Variational Formulas for L_p,q-Mixed Volumes

5. The L_p,q-Mixed Volumes

6. The L_p,q-Minkowski Problems

7.1. L_p,q-Mixed Affine Surface Areas

7.2. L_p,q-Mixed Geominimal Surface Area

7.3. L_p,q-John Ellipsoids