The geominimal integral curvature

1.   Introduction

The geominimal surface area belongs to the research category of convex geometric analysis. The classical geominimal surface area, which can be dated back to 1974, was firstly introduced by Petty ^[21], in his paper, the classical Petty body was obtained and some classical affine isoperimetric inequalities were established. With the development of the $L_p$-Brunn-Minkowski theory, the classical geominimal surface area has been extended to $L_p$ cases by Lutwak ^[14] (for $p > 1$) and Ye ^[26] (for $-n\neq p < 1$) (also see ^[28,30]). The dual $L_p$-Brunn-Minkowski theory was introduced by Lutwak ^[13,15]. Wang and Chen ^[25] introduced the dual $L_p$ geominimal surface area. For more results of the geominimal surface area, one can refer to ^{[9,10,11,18,27,29,31]} and so on.

We shall work in the $n$-dimensional Euclidean space $\mathbb{R}^n$ with the standard inner product $x\cdot y$ of $x$ and $y$ in $\mathbb{R}^n$. For $x\in\mathbb{R}^n$, write $|x| = \sqrt{x\cdot x}$ for the Euclidean norm of $x$. We call a set $K\subset \mathbb{R}^n$ is convex if provided that for any two points $x, y\in K$ and $\lambda \in [0, 1]$, one has $\lambda x+(1-\lambda)y\in K$. A convex subset $K\subset \mathbb{R}^n$ is called a convex body if $K$ is compact with nonempty interior, and the interior point of a convex body $K$ can be written as $\mathrm{int}K$. Moreover, if for any two points $x, y\in K$ ($x\neq y$) and $\lambda \in (0, 1)$, $\lambda x+(1-\lambda)y\in \mathrm{int}K$ holds, we call the convex body $K$ is a strictly convex.

Throughout this paper, let $\mathcal{K}^n$ be the class of all convex bodies in $\mathbb{R}^n$ and $\mathcal{K}_0^n$ be the class of all convex bodies in $\mathbb{R}^n$ that contain the origin in their interiors and $\mathcal{K}_e^n$ be the class of all origin-symmetric convex bodies in $\mathbb{R}^n$. The standard Lebesgue measure of a set $K$ in $\mathbb{R}^n$ will be denoted by $|K|$, we call $|K|$ is the $n$-dimensional volume of a set $K\in \mathcal{K}^n$. The volume radius of the convex body K is defined as $vrad(K) = (|K|/\omega_n)^{1/n}$. The origin-symmetric unit ball in $n$-dimensional Euclidean space is denote by $B^n_2$, i.e., $B^n_2 = \{x\in \mathbb{R}^n: |x|\leq1\}$, and we use $\omega_n$ to denote the volume of the unit ball $B_2^n$. We use $S^{n-1}$ to denote the unit sphere in $\mathbb{R}^n$, and we let $\kappa_n$ be the surface area measure of $B^n_2$. Let $K|{u^\bot}$ be the orthogonal projection of $K$ onto the subspace orthogonal to $u$ and $V_{n-1}(K|{u^\bot})$ be the $(n-1)$-dimensional volume of $K|{u^\bot}$. Let $C(S^{n-1})$ be the set of continuous functions defined on the unit sphere $S^{n-1}$.

As we all know, the notion of geominimal surface area was introduced by Petty ^[21]. For $K\in\mathcal{K}_0^n$, the geominimal surface area of $K$, is defined by the following optimal problem:

Here $Q^*$ denotes polar body of $Q$, $h_Q(\cdot)$ is the support function of convex body $Q$ (see Section 2), $\mathcal{K}_s^n$ is the class of convex bodies in $\mathbb{R}^n$ whose Santal$\mathrm{\acute{o}}$ points are at origin point $o$ (see ^[23]), and $S(K, \cdot)$ is the surface area measure of convex body $K$, i.e., for any $K\in \mathcal{K}^n$ and any measurable subset $\Omega\subset S^{n-1}$, the surface area measure $S(K, \Omega)$, is defined by

where $\nu_K^{-1}:S^{n-1}\rightarrow \partial K$ is the inverse Gauss map and $\mathcal{H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure on $\partial K$.

In ^[16], Lutwak extended over body $Q$ restricted to $\mathcal{K}_{0}^n$. It follows that

For any two convex bodies $K, Q\in\mathcal{K}^n$, the mixed volume $V_1(K, Q)$ of $K, Q$ is defined by

Moreover, for any $K\in\mathcal{K}^n$, the volume of $K$ is defined by

Together with (1.2), one has,

Lutwak ^[16] proved the uniqueness of the solution to (1.4). This shows that the mixed volume $V_1(\cdot, \cdot)$ is a necessary premise of classical geominimal surface area. Hence, utilizing the relationship between the minimum surface area and the corresponding mixed volume, one can investigate the geominimal integral curvature with the help of the mixed entropy.

We define a new concept of the entropy functional of the convex body, i.e., the entropy functional $\mathcal{E}(K)$ of the convex body $K\in\mathcal{K}_0^n$ can be defined by

where the integration is with respect to spherical Lebesgue measure (see, Huang et al.^[7]).

For two convex bodies $K, L\in\mathcal{K}_0^n$, the $Lp$ mixed entropy is defined by

where $K\hat{+}_p\varepsilon \diamond L$ is the harmonic $L_p$ combination of $K, L$ for $p\geq1$ and $K\hat{+}_p\varepsilon \diamond L\in\mathcal{K}_0^n$ is a new convex body. In Section 2, we will show that the convex body $K\hat{+}_p\varepsilon \diamond L\in\mathcal{K}_0^n$ can be defined for all $p\in\mathbb{R}$ and even for negative $\epsilon$ of sufficiently small absolute value. In this paper, we mainly study the mixed entropy functional when $p = 0$, i.e., for $K, L\in\mathcal{K}_0^n$

where $\rho_L$ is the radial function of convex body $L$ and $J(K, \cdot)$ is the Aleksandrov integral curvature of the convex body $K$ (see Section 2). For some of the the Aleksandrov integral recent resilts, see ^{[1,3,8,19,20,22]}.

There exists a natural problem: whether there is a convex body $L\in\mathcal{K}_0^n$ with $|L^*| = \omega_n$ such that $L$ is a solution to the following problem:

In this paper, let ${G}_\mathcal{E}(K)$ be the geominimal integral curvature of the convex body $K\in\mathcal{K}_0^n$, it can be defined by optimal problem in (1.6). Together with (1.5), it is equivalent to solve the optimal problem as follows:

Based on the concept of the Petty body, we will study the entropy form of the Petty body. The following is our main result.

Theorem 1.1. Let $K\in \mathcal{K}^n_o$, then there exists a convex body $M\in \mathcal{K}^n_e$ with $|M^*| = \omega_n$ such that

In addition, in the plane $\mathbb{R}^2$, these bodies are unique or their polar bodies are parallelograms with parallel sides.

This paper is organized as follows. In Section 2, we collect some basic concepts and various facts that will be used in the proofs of our results. In Section 3, we study the properties of the $L_0$ mixed entropy. In Section 4, we obtain the existence and uniqueness of the geominimal integral curvature.

2.   Background and notation

2.1.   Convex bodies

We now introduce some basic facts and standard notations needed in this paper. For more details and concepts in convex geometry, please see ^[4,5,24].

The Minkowski sum of two convex sets $K$ and $L$ is denoted by

The scalar product of $\lambda\in\mathbb{R}$ and $K\in\mathcal{K}^n$ is defined by $\lambda K = \{\lambda x:x\in K\}$. For any convex body $K$ in $\mathbb{R}^n$, the support function of $K$, $h_K: {S^{n-1}} \rightarrow \mathbb{R}$, is defined by

For the convex body $K$ and $u\in {S^{n-1}}$, the hyperplane

is called the supporting hyperplane of $K$ with unit normal $u$. For $x\in \partial K$, if there is only one supporting hyperplane of $K$ passing through point $x$, we call $x\in\partial K$ is a smooth point. If there exist more than a supporting hyperplane of $K$ passing through point $x$, we call $x\in\partial K$ is a singlar point. For each $x\in \partial K$ is a smooth point, we call the convex body $K$ is smooth. If a convex body $K$ is strictly convex and smooth, we say that $K$ is a regular convex body. For each $K\in \mathcal{K}^n$, there exists a regular convex body sequence $K_i$ such that $K_i$ converges to $K$ as $i\rightarrow \infty$ (see ^[24]).

Let $l_u = \{tu: t\geq 0\}$ for $u\in {S^{n-1}}$. If $L\cap l_u$ is a closed line segment for all $u\in S^{n-1}$, we say $L\subset {\mathbb{R}}^n$ is star-shaped with respect to the origin. Let $L$ be compact and star-shaped with respect to the origin, the radial function $\rho_L: {S^{n-1}}\rightarrow [0, \infty)$ is defined by

A compact star-shaped set with respect to the origin is uniquely determined by its radial function. If $\rho_L$ is positive and continuous on ${S^{n-1}}$, then the star-shaped $L$ is called a star body about the origin. Let $\mathcal{I}_0^n$ be the set of all star bodies about the origin. Clearly, the radial function of a convex body in $\mathcal{K}_0^n$ is continuous and positive, i.e., $\mathcal{K}_0^n\subset\mathcal{I}_0^n$. If $K\in\mathcal{K}_0^n$, then

And the volume of $K\in\mathcal{K}_0^n$ can be rewritten by

For Borel set $\eta \subset S^{n-1}$, let $\rho:\eta\rightarrow (0, \infty)$ be a continuous function, then the set $\{\rho(u)u:u\in \eta\}$ is a Borel set. For a set $K\subset\mathbb{R}^n$, the convex hull of $K$, write conv$K$, is the intersection of all convex sets containing $K$. Hence, the convex hull $\langle\rho\rangle$ generated by $\rho$,

is a compact set (see ^[24]). Let $\eta\subset S^{n-1}$ always be a closed set and not contained in any great hemisphere of $S^{n-1}$, then we have $\langle\rho\rangle\in \mathcal{K}^n_0$. By the definition of convex hull of the function $\rho$, we have

If the function $\rho$ is an even function on $S^{n-1}$, then the convex hull $conv\{\rho(u)u:u\in S^{n-1}\}$ is an origin-symmetric body. Thus a convex body $K$ is origin-symmetric if and only if the radial function $\rho_K$ of the convex body $K$ is an even function.

The definitions of radial function and support function immediately give that for $\lambda > 0$ and $K, L\in\mathcal{K}_0^n$, one has $h_{\lambda K} = \lambda h_K$, $\rho_{\lambda K} = \lambda \rho_K$ and $h_{K+L} = h_K+h_L$. Moreover

If $K\in \mathcal{K}^n_0$, the following formulas hold for all $u\in S^{n-1}$,

where $K^*$ is the polar body of $K$, and it is given by

On the set $\mathcal{K}^n$, we consider the topology generated by the Hausdorff metric $d_H(\cdot, \cdot)$. For $K, K'\in \mathcal{K}^n$, the Hausdorff metric $d_H(K, K')$ is defined by

A sequence $\{K_i\}_{i\geq 1}\subset \mathcal{K}^n$ converges to a convex body $K_0\in \mathcal{K}^n$ if $d_H(K_i, K_0)\rightarrow 0$ as $i\rightarrow \infty$, i.e.,

The radial metric is defined by

for $K, L\in\mathcal{I}_0^n$. We use the fact that on $\mathcal{K}_0^n$, the Hausdorrf metric and the radial metric are topologically equivalent, i.e.,

for $\{K_i\}_{i\geq1}\subset\mathcal{K}_0^n$ and $K_0\in\mathcal{K}_0^n$.

If the function $f:\eta\rightarrow (0, \infty)$ is continuous, the Wulff shape $[f]\in\mathcal{K}^n_0$ determined by $f$ is a convex body defined by

Note that, if $f = h_K$ is a support function of convex body $K\in \mathcal{K}_0^n$, one has

Furthermore, if the function $f:\eta\rightarrow (0, \infty)$ is continuous, we have (see e.g., ^[17] p.95)

The $L_p$ Minkowski combination is a basic concept in the $L_p$-Brunn-Minkowski theory. For each $p\geq1$, the Minkowski-Firey $L_p$-combination $K+_pL$ introduced by Firey (see, e.g., ^[24]) can be defined by the support function as follows, i.e., for $K, L\in \mathcal{K}_0^n$ and $a, b > 0$

Now we fix $p\neq 0.$ For $K, L\in \mathcal{K}_0^n$ and $a, b > 0$, we define the general $L_p$ Minkowski combination, $a\cdot K+_pb \cdot L\in \mathcal{K}_0^n$, via the Wulff shape,

When $p = 0,$ we define $a\cdot K+_0b \cdot L\in \mathcal{K}_0^n$ via the Wulff shape,

Note that "$\cdot$" is written without its subscript $p$.

For any $p\in\mathbb{R}$, Huang et al.^[7] gave the definition of the $L_p$-harmonic combination $(1-\lambda)\diamond K\widehat{+}_p \lambda \diamond L\in \mathcal{K}_0^n$, i.e.,

Hence, together with (2.4)–(2.6), we obtain that

Let $\{K_i\}_{i = 1}^\infty\subset\mathcal{K}^n_0$ and $K\in\mathcal{K}^n_0$, combined with (2.4), this implies that

2.2.   The integral curvature

For a convex body $K$ in $\mathbb{R}^n$, the $Gauss\ image$ of $\sigma\subset\partial K$ is defined by

The $reverse\ Gauss\ image$ of $\eta\subset S^{n-1}$ is defined by

Let $\sigma_K = \{x: x\in\partial K \ \mathrm{is \ a\ singlar \ point}\}\subset\partial K$. It is known that $\mathcal{H}^{n-1}(\sigma_K) = 0$ (see, p.84 of Schneider ^[24]). The $Gauss \ map$ of the convex body $K$ is defined by

From Lemma 2.2.12 of Schneider ^[24] we know that the Gauss map $\nu_K$ is continuous. The set $\eta_K\subset S^{n-1}$ consisting of all $v\in S^{n-1}$, for which the set $\pmb{\nu}_K^*(\{v\})$, abbreviated as $\pmb{\nu}_K^*(v)$, contains more than a single element, is of $\mathcal{H}^{n-1}$-measure 0. The $inverse\ Gauss \ map$ of the convex body $K$ is defined by

From Lemma 2.2.12 of Schneider ^[24] we also know that the function $\nu_K^{-1}$ is continuous.

For $K\in\mathcal{K}_0^n$, define the radial map of the convex body $K$

for $u\in S^{n-1}$. Note that $r_K^{-1}(\cdot):\partial K\rightarrow S^{n-1}$ is the map $r_K^{-1}(x) = \bar{x} = x/|x|$. For $\omega\subset S^{n-1}$, define the $radial\ Gauss\ image$ of $\omega$ by

Thus, for $u\in S^{n-1}$

Define the $radial\ Gauss\ map$ of the convex body $K\in\mathcal{K}_0^n$

where $\omega_K = \{x/{|x|}:x\in \sigma_K\}$.

Define the $reverse\ radial\ Gauss\ image$ of $\eta\subset S^{n-1}$ by

The $inverse\ radial\ Gauss\ map$ of the convex body $K\in\mathcal{K}_0^n$ is defined by

Note that since both $r_K^{-1}$ and $\nu_K^{-1}$ are continuous, $\alpha^{-1}_K$ is continuous.

The integral curvature $J(K, \cdot)$ of convex body $K\in \mathcal{K}^n_0$ is defined by,

for each Borel set $\omega\subset S^{n-1}$. The total integral curvature of convex body $K$, is the surface area of the unit sphere $S^{n-1}$, thus $J(K, S^{n-1}) = \kappa_n$. The concept of integral curvature was introduced by Aleksandrov.

Following formula ($2.10$), and characteristic function $I$ on $S^{n-1}$, then

the last identity holds from the fact that $v\in \pmb\alpha_K(\omega)$ if and only if $\alpha_K^{-1}\in\omega$ for almost all $u$ with respect to the spherical Lebesgue measure (see (2.20) in ^[6]). Furthermore, from formula (2.11), we have that

for each continuous function $f$ on $S^{n-1}$.

Lemma 2.1. (^[6] Lemma 2.2) Let $K_i\in \mathcal{K}^n_0$ be such that $\lim_{i\rightarrow \infty}K_i = K_0\in\mathcal{K}^n_0$. Let $\omega = \bigcup_{i = 0}^\infty\omega_{K_i}$ be the set $\mathrm{(}$of $\mathcal{H}^{n-1}$-measure zero$\mathrm{)}$ off of which all of the $\alpha_{K_i}$ are defined. If $u_i\in S^{n-1}\backslash\omega$ are such that $\lim_{i\rightarrow \infty}u_i = u_0\in S^{n-1}\backslash \omega$, then $\lim_{i\rightarrow \infty}\alpha_{K_i}(u_i) = \alpha_{K_0}(u_0).$

Lemma 2.2. (^[6] Lemma 2.5) Let $K\in \mathcal{K}^n_0$, then

for each $\eta\subset S^{n-1}$.

Note that if $K_i\rightarrow K_0$ in the Hausdorff metric, then for all $f\in C(S^{n-1})$, by formulas (2.9), (2.12), Lemmas 2.1 and 2.2, one has

This proves that the integral curvature $J(K, \cdot)$ is weakly convergence measure.

Lemma 2.3. (^[12] Lemma 2.1) If a sequence of measures $\{\mu_i\}_{i = 1}^\infty$ on $S^{n-1}$ converges weakly to a finite measure $\mu$ on $S^{n-1}$ and a sequence of functions $\{f_i\}_{i\geq 1}\subset C(S^{n-1})$ converges uniformly to a function $f_0\in C(S^{n-1})$, then

Thus, by Lemma 2.3, if $\{f_i\}_{i\geq 1}\subset C(S^{n-1})$ is uniformly convergent to $f_0\in C(S^{n-1})$ and $\{K_i\}_{i\geq 1}\subset \mathcal{K}_0^n$ converges to $K_0\in \mathcal{K}_0^n$ in the Hausdorff metric, together with the weak convergence of $J(K, \cdot)$, we have

The Blaschke selection theorem is a powerful tool in convex geometry (see ^[5,24]) and will be often used in this paper. It reads: Every bounded sequence of convex bodies has a subsequence that converges to a convex set.

We will also use the following lemmas in the proofs of our main results.

Lemma 2.4. (see ^[12]) If $\{K_i\}_{i\geq 1}\subset \mathcal{K}_0^n$ is a bounded sequence and $\{|K^*_i|\}_{i\geq 1}$ is also a bounded sequence, there is a subsequence $\{K_{i_j}\}_{j\geq 1}$ of the sequence $\{K_i\}_{i\geq 1}$ and a body $K\in\mathcal{K}_0^n$ such that $K_{i_j}\rightarrow K.$ In addition, if $|K^*_i| = \omega_n$, then $|K^*| = \omega_n$.

Lemma 2.5. (see ^[16]) Let $\{K_i\}_{i\geq 1}\subset \mathcal{K}_0^n$ be a convergent sequence with limit $K_0$, i.e., $K_i\rightarrow K_0$ in the Hausdorff distance. If the sequence$\{|K_i^*|\}_{i\geq 1}$ is bounded, then $K_0\in\mathcal{K}_0^n$.

3.   Properties for the $L_0$ mixed entropy $\mathcal{E}_0(K, L)$

In this section, we mainly prove some properties for the $L_0$ mixed entropy. We now prove the continuity of the $L_0$ mixed entropy as follows.

Proposition 3.1. Let ${\left\{K_i\right\}}_{i = 0}^{\infty}\subset \mathcal{K}^n_0$ and ${\left\{L_i\right\}}_{i = 0}^{\infty}\subset \mathcal{K}^n_0$ be two sequences of convex bodies such that $K_i\rightarrow K_0\in \mathcal{K}^n_0$ and $L_i\rightarrow L_0\in \mathcal{K}^n_0$ as $i\rightarrow \infty$ in the Hausdorff metric, then

Proof. Since $K_i\rightarrow K_0\in \mathcal{K}^n_0$ and $L_i\rightarrow L_0\in \mathcal{K}^n_0$ as $i\rightarrow \infty$ in the Hausdorff metric, then $J(K_i, \cdot)$ converges weakly to $J(K_0, \cdot)$ and the radial functions $\rho_{L_i}$ converges uniformly to $\rho_{L_0}$, as $i\rightarrow \infty$. And there are two constants $r, \ R > 0$ such that for all $i\geq1$,

We have $r\leq \rho_{L_i}\leq R$, for all $i\geq1$. Furthermore, together with the continuity of the logarithmic function on $[r, R]$, we get

By formula (2.13), one has

Proposition 3.2. Let ${\left\{K_i\right\}}_{i = 1}^{\infty}\subset \mathcal{K}^n_0$ and $K\in\mathcal{K}^n_0$ be regular convex bodies such that $K_i\rightarrow K$ as $i\rightarrow \infty$ in the Hausdorff metric. For ${\left\{L_i\right\}}_{i = 1}^{\infty}\subset \mathcal{K}^n_e$, then $\{\mathcal{E}_0(K_i, L_i)\}^\infty_{i = 1}$ is bounded and ${\left\{L_i\right\}}_{i = 1}^{\infty}$ is uniformly bounded if and only if there exist $\alpha, r > 0$ such that for all $i\geq 1$

Proof. The boundedness of $\{\mathcal{E}_0(K_i, L_i)\}^\infty_{i = 1}$ is equivalent to the boundedness of $\{-\mathcal{E}_0(K_i, L_i)\}^\infty_{i = 1}$, which shows that there are constants $c$ and $C$ such that $c\leq -\mathcal{E}_0(K_i, L_i)\leq C$ for all $i\geq 1$.

We first show that the sequence $\{L_i\}_{i = 1}^\infty$ contains a small ball. For $u_i\in S^{n-1}$, let

Since the sequence $\{L_i\}_{i = 1}^\infty$ is bounded, there is a constant $\beta > 0$, such that $R_i(u_i)\leq\beta$ for all $i$, we have $\beta^{-1}B^n_2\subset L_i^*$ for all $i$. Hence

for all $i$.

On the other hand, by the Blaschke selection theorem, there is a subsequence of ${\left\{L_i\right\}}_{i = 1}^{\infty}$, for convenience, we still record it as ${\left\{L_i\right\}}_{i = 1}^{\infty}$, and a compact convex set $L_0$, such that $L_i\rightarrow L_0$ as $i\rightarrow \infty$ in the Hausdorff metric, the radial function sequence $\rho_{L_i}$ is uniformly continuous, we have that $\rho_{L_0}$ is continuous, now we prove that $L_0$ contains a small ball $rB^n_2$, if not, then there is a nonzero set $\omega$ and a sufficiently small real $\epsilon > 0$ such that $\omega = \{u:\rho_{L_0}(u) < \epsilon\}$, according to the regularity of the convex body $K$, we have $J(K, \omega) > 0$ and $J(K, S^{n-1}\backslash\omega)\log R < \infty$. Thus, by Proposition 3.1,

Let $\epsilon\rightarrow 0^+$, hence $J(K, \omega)\log \epsilon\rightarrow -\infty$, This is a contradiction to the boundedness of the mixed entropy $\mathcal{E}_0(K_i, L_i)$.

Now we prove that the sequence $\{L_i\}_{i = 1}^\infty$ is bounded. We let $R_i(u_i) = \max\{\rho_{L_i}(u):u\in S^{n-1}\}$ for some $u_i\in S^{n-1}$. Since the sequence ${\left\{L_i\right\}}_{i = 1}^{\infty}$ contains a small ball, there is a constant $r > 0$ such that $rB^n_2\subset L_i$ for all $i\geq1$, let $Q_i$ be the convex hull of point $R_i(u_i)u_i$ and $rB_2^n|{u^\perp_i}$, i.e.,

Obviously, $Q_i\subset L_i$, together with the monotonicity of volume and (1.3),

By the Blaschke-Stantal$\mathrm{\acute{o}}$ inequality, i.e., for $L_i\in \mathcal{K}_e^n$,

Combined with $|L_i^*|\geq \alpha$, this implies that

for all $i$. Thus the sequence $\{L_i\}_{i = 1}^\infty$ is bounded. There are two constants $r, R > 0$ such that $r B^n_2\subset L_i\subset R B^n_2$ for all $i$, together with (2.10), we know that $J(K_i, S^{n-1}) = \kappa_n$, then for all $i$,

This shows that the sequence $\{\mathcal{E}_0(K_i, L_i)\}^\infty_{i = 1}$ is bounded.

Remark 1. According to the above proof of Proposition 3.2, if ${\left\{L_i\right\}}_{i = 1}^{\infty}\subset \mathcal{K}^n_e$ and some $\alpha > 0$ such that $rB^n_2\subset L_i \ \mathrm{and}\ |L_i^*|\geq \alpha$, then we can remove the condition that $K\in \mathcal{K}^n_0$ is a regular convex body, we also obtain the results that $\{\mathcal{E}_0(K_i, L_i)\}^\infty_{i = 1}$ is bounded and ${\left\{L_i\right\}}_{i = 1}^{\infty}$ is uniformly bounded.

4.   The geominimal integral curvature

Throughout this section, we suppose that $K\in \mathcal{K}_0^n$, we mainly prove the existence and uniqueness of the Entropy-Petty body. For further discussion, we introduce the continuity of the geominimal integral curvature ${G}_\mathcal{E}(K)$. We first define the geominimal integral curvature ${G}_\mathcal{E}(K)$ as follows:

Definition 4.1. Suppose $K\in \mathcal{K}_0^n$ is a convex body, the geominimal integral curvature of $K$ is defined by

Remark 2. We show that the above definition is well defined. In fact, since $|L^*| = \omega_n$ and $L\in\mathcal{K}_e^n$, by the Blaschke-Stantal$\mathrm{\acute{o}}$ inequality, we have $|L|\leq\omega_n$. Hence, by (2.1) and the Jensen inequality

Hence $\int_{S^{n-1}}\rho_L(u)du$ is uniformly bounded. In the next, we assume $K\in\mathcal{K}_0^n$ is a regular convex body, by the concavity of Logarithmic function, we obtain that

For any $K\in \mathcal{K}_0^n$, we choose a regular convex body sequence $K_i$ such that $K_i\rightarrow K$ as $i\rightarrow \infty$, combined with Proposition 3.1, this implies that

Hence Definition 4.1 is well defined.

In the next, we prove our mainly result Theorem 1.1.

proof of Theorem 1.1. Firstly, we prove the existence. By Definition 4.1, there is a sequence $\{M_i\}_{i = 1}^\infty\subset\mathcal{K}_e^n \mathrm{\ with \ }|M_i^*| = \omega_n$ such that

Let $K\in\mathcal{K}_0^n$ be a regular convex body, we have

By formula (2.4), we get

Let $R_i(u_i) = \max\{\rho_{M^*_i}(u):u\in S^{n-1}\}$, and since $M_i\in\mathcal{K}_e^n$, we have $[-R_i(u_i)u_i, R_i(u_i)u_i]\subset M_i^*.$ Hence $h(M_i^*, u)\geq R_i(u_i)|u\cdot u_i|$ for all $u\in S^{n-1}$. Therefore

Now, assume $K\in \mathcal{K}_0^n$ is not a regular convex body, we can choose a regular convex body sequence $K_i$ such that $K_i\rightarrow K$ as $i\rightarrow \infty$, combined with Proposition 3.1, this implies that

Since the integral on the left is independent of $u_i$, this implies that $R_i(u_i)$ is uniformly bounded. Hence, there exists $r > 0$ such that $rB^n_2\subset M_i$ for all $i\geq1$. By Proposition 3.2, the sequence $\{M_i\}_{i = 1}^\infty$ is bounded. By the Blaschke selection theorem, there is a subsequence, for convenience, it is still recorded it as $\{M_i\}_{i = 1}^\infty$, which converges to a compact convex set $M$. Since $|M^*_{i}| = \omega_n$, by Lemma 2.5, we have $M\in\mathcal{K}_0^n$. Therefore $M_i\in \mathcal{K}^n_e$, gives $M\in\mathcal{K}_e^n$. By Proposition 3.1, we obtain that

Next, we prove the uniqueness of theorem in the plane $\mathbb{R}^2$. Assume that there are two convex bodies $M_1, M_2\in\mathcal{K}_e^2$ with $|M_1^*| = |M_2^*| = \pi$ such that

Now we define a new set $M\subset \mathbb{R}^n$ about $M_1, M_2$ and together with (2.7), we have

Combining (2.3) and (2.8), we obtain $M = \langle\rho^{1/2}_{M_1}\rho^{1/2}_{M_2}\rangle$, this together with $M_1, M_2\in\mathcal{K}_e^2$ implies that the function $\rho(u) = \rho^{1/2}_{M_1}(u)\rho^{1/2}_{M_2}(u)$ is even function on $S^{n-1}$. Hence $M\in\mathcal{K}_e^2$ and

Furthermore, by the log Brunn-Minkowski inequality in the plane (see ^[2]),

with equality if and only if $M_1^*$ and $M_2^*$ are dilates or they are parallelograms with parallel sides. If $M_1^*$ and $M_2^*$ are dilates, we let $M_1^* = sM_2^*$ for real number $s > 0$ and together with $|M_1^*| = s^2|M_2^*|$, we have $s = 1$, thus we see $M_1 = M_2$, which can be checked that $\mathrm{vrad}(M^*)\geq1$, with equality if and only if $M_1 = M_2$ or their polar bodies are parallelograms with parallel sides. By (4.1) and Definition 4.1, we have

Hence, this forces $\mathrm{vrad}(M^*) = 1$ and then $M_1 = M_2$ or their polar bodies are parallelograms with parallel sides.

We will prove the continuity of ${G}_\mathcal{{E}}(K)$ as follows:

Theorem 4.1. Let $\{K_i\}_{i = 1}^\infty\subset\mathcal{K}^n_0$ and $K\in \mathcal{K}_0^n$ be such that $K_i\rightarrow K$ as $i\rightarrow \infty$ in the Hausdorff metric, then $\lim_{i\rightarrow \infty}{G}_\mathcal{E}(K_i) = {G}_\mathcal{E}(K).$

Proof. Let $\{K_i\}_{i = 1}^\infty\subset\mathcal{K}_0^n$ be such that $K_i\rightarrow K\in\mathcal{K}_0^n$ as $i\rightarrow \infty$. For any fixed small $\varepsilon > 0$, by Definition 4.1 and Proposition 3.1, there is a convex body $M_\varepsilon$ with $|M_\varepsilon^*| = \omega_n$, we have

Since $\varepsilon > 0$ is arbitrary small, one has

We now assume that $M_i\in\mathcal{K}_e^n$ with $|M_i^*| = \omega_n$ such that ${G}_\mathcal{E}(K_i) = -\mathcal{E}_0(K_i, M_i)$,

Since both the sequence $\mathcal{E}_0(K_i, M_i)$ and $\{M_i\}_{i = 1}^\infty$ are bounded, the Blaschke selection theorem now yields that a subsequence $\{M_{i_j}\}_{j = 1}^\infty$ of $\{M_i\}_{i = 1}^\infty$ converges to some compact convex set $M^\prime$. But $|M_{i_j}^*| = \omega_n$, by LemmaS 2.4, 2.5, and $M_i\in\mathcal{K}_e^n$, the set $M^\prime\in\mathcal{K}_e^n$ is an origin-symmetric convex body and $|(M^\prime)^*| = \omega_n$. Together with the Definition 4.1, Proposition 3.1 and Theorem 1.1, we obtain

Combining (4.2) and (4.3), we complete the proof, i.e.,

In the following corollary, we show that if the convex body $K$ is an origin-symmetric polytope, then the optimal problem has an origin-symmetric polytope solution.

Corollary 4.1. Let $K\in\mathcal{K}_e^n$ be a polytope with vertices $u_1, u_2, \cdot\cdot\cdot, u_m$. If $M\in\mathcal{K}_e^n$ such that

then $M$ is a polytope with vertices $v_1, v_2, \cdot\cdot\cdot, v_m$. Moreover, $v_i = \lambda_iu_i$ for $\lambda_i > 0$, $i\in\{1, 2, \cdot\cdot\cdot, m\}$.

Proof. Let $K\in\mathcal{K}_e^n$ be a polytope with $\mathrm{vert}K = \{u_1, u_2, \cdot\cdot\cdot, u_m\}$ ($m = 2N\geq n+1$). Obviously, $\{\frac{u_1}{|u_1|}, \cdot\cdot\cdot, \frac{u_m}{|u_m|}\}\subset S^{n-1}$ are not concentrated in any closed hemisphere of $S^{n-1}$. Then the integral curvature measure $J(K, \cdot)$ about convex body $K$ is the discrete measure concentrated on $\{\frac{u_1}{|u_1|}, \cdot\cdot\cdot, \frac{u_m}{|u_m|}\}\subset S^{n-1}$. Let $P$ be a polytope

where $\bar{u}_i = \frac{u_i}{|u_i|}\in S^{n-1}$ for $i = 1, ...m$. Let $u_{P, i} = \rho(M, \bar{u}_i)\bar{u}_i\in \partial P$ be vertices of polytope $P$, then these are $\lambda_i > 0$ such that $u_{P, i} = \lambda_iu_i$ for $i\in\{1, 2, \cdot\cdot\cdot, m\}$.

In the next, we only need prove $P = M$. By (4.4), we have $\rho(P, \bar{u}_i) = \rho(M, \bar{u}_i)$ $(1\leq i\leq m)$ and $P\subset M$. Thus

We obtain

This shows that $vrad(P^*) = vrad(M^*) = 1$. Hence we know that $|M| = |P|$. Thus $P = M$.

5.   Conclusions

In this paper, the geominimal integral curvature on the convex body is introduced. The existence and uniqueness of the geominimal integral curvature are proved. Some other properties for the geominimal integral curvature, such as continuity, are investigated.

Conflict of interest

The author declares that they have no competing interests.

Acknowledgments

This research is supported by Youth Thousand Talents Plan, Grant (No. GK202101008) and Postgraduate Innovation Team Project of Shaanxi Normal University (No. TD2020008Z).