The curvature entropy inequalities of convex bodies

Deyan Zhang

doi:10.1515/math-2024-0066

Article Open Access

The curvature entropy inequalities of convex bodies

Deyan Zhang

Published/Copyright: October 18, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

There are many entropy inequalities in geometry, some of them can be seen as the Minkowski inequalities in the form of entropy, which play important roles in convex geometry. In this article, let P and Q be convex bodies, we introduce the mixed curvature entropy M f ( P , Q ) and the curvature entropy E f ( P , Q ) with respect to a continuous function f . Moreover, we establish the log -Minkowski inequalities of the mixed curvature entropy and the curvature entropy for two classes of three-dimensional convex bodies.

Keywords: log-Minkowski inequality; mixed curvature entropy; curvature entropy

MSC 2010: 53A40; 52A20

1 Introduction

Let K be a bounded closed set in the n -dimensional Euclidean space R n , if it is convex and has non-empty interior, then it is called a convex body. Throughout the article, C n and C o n denote the sets of all convex bodies and all convex bodies containing the origin in their interiors, respectively. Let P , Q ∈ C n . For any real number μ ∈ ( 0 , 1 ) , ( 1 − μ ) P + μ Q is the Minkowski combination of P and Q , and

( 1 − μ ) P + μ Q = { ( 1 − μ ) p + μ q ∣ p ∈ P , q ∈ Q } .

Geometric inequalities are the central objects of interest in convex geometry, such as all kinds of volume inequalities. One of the most important volume inequalities in convex geometry is the classical Brunn-Minkowski inequality, which asserts that

(1.1) V ( ( 1 − μ ) P + μ Q ) 1 n ≥ ( 1 − μ ) V ( P ) 1 n + μ V ( Q ) 1 n ,

where the equality holds if and only if P and Q are homothetic, this also implies the log -concavity of the volume measure. Many of its generalizations, improvements, and applications are discovered, see Gardner’s article [1].

In 1962, Firey [2] introduced the L p -Minkowski combination of convex bodies for p ∈ [ 1 , + ∞ ) , which generalized the classic Minkowski combination. In the 1990s, Lutwak [3,4] extended many classical results to the case of L p -Minkowski combination and thus established a more general theory. At the same time, the L p -Brunn-Minkowski inequality was built as a more general version of inequality (1.1). However, it is very difficult for p ∈ [ 0 , 1 ) . Particularly, when p equals zero, the corresponding inequality is called the log -Brunn-Minkowski inequality which is the strongest among all cases of p ∈ ( 0 , + ∞ ) . If P , Q ∈ C 2 and are symmetric about the origin, Böröczky et al. [5] proved the log -Brunn-Minkowski inequality:

(1.2) V ( ( 1 − μ ) P + o μ Q ) ≥ V ( P ) ( 1 − μ ) V ( Q ) μ ,

and the equality holds if and only if P and Q are parallelograms with parallel sides or P and Q are dilates.

Let h P and h Q be the support functions of P and Q , respectively, Böröczky et al. proved that inequality (1.2) can be equivalently transformed into the following log -Minkowski inequality:

(1.3) ∫ S 1 log h Q h P d V ¯ P ≥ 1 2 log V ( Q ) V ( P ) ,

where d V ¯ P represents the cone-volume probability measure of P .

Inequality (1.3) and its generalizations are getting more and more attention [6–9]. For example, Yang and Zhang [9] proved that inequality (1.3) holds for some three-dimensional convex bodies. Besides that, there are many versions of inequality (1.3) in the form of entropy, see [10–12]. For instance, Ma et al. [13] defined the curvature entropy E ( P , Q ) for P , Q ∈ C o n by

(1.4) E ( P , Q ) = ∫ S n − 1 log H n − 1 ( P ) H n − 1 ( Q ) d V P ,

where d V P denotes the cone-volume measure of P and H n − 1 ( P ) denotes the Gauss curvature of the boundary ∂ P . Furthermore, if P and Q are planar convex bodies and P is symmetric about the origin, they proved the following inequality:

(1.5) E ( P , Q ) ≤ V ( P ) 2 log V ( Q ) V ( P ) .

It follows from (1.4) that E ( P , Q ) can be viewed as a curvature entropy under the logarithmic function. In view of this, in the present article, we generalize the definition of curvature entropy to a more general case for any continuous function. That is, assuming f is a continuous function, we can define the mixed curvature entropy M f ( P , Q ) and the curvature entropy E f ( P , Q ) with respect to f by

M f ( P , Q ) = ∫ S n − 1 f H n − 1 ( P ) H n − 1 ( Q ) d V P , Q

and

E f ( P , Q ) = ∫ S n − 1 f H n − 1 ( P ) H n − 1 ( Q ) d V P .

It can be easily verified that E f ( P , Q ) is exactly E ( P , Q ) when f is the logarithmic function.

For the mixed curvature entropy, in Section 3, we discuss its fundamental properties. Particularly, if f = x p , the mixed curvature entropy M f ( P , Q ) (or M p ( P , Q ) ) and its normalization M ¯ f ( P , Q ) (or M ¯ p ( P , Q ) , are interesting. Let M ¯ 0 ( P , Q ) = lim p → 0 M ¯ p ( P , Q ) , considering the two classes of convex bodies by Yang and Zhang [9], we establish the mixed curvature entropy inequalities.

In Section 4, we define the general curvature entropy E f ( P , Q ) . When f = x p , we discuss some fundamental properties of the mixed curvature entropy E f ( P , Q ) (or E p ( P , Q ) ) and its normalization E ¯ f ( P , Q ) (or E ¯ p ( P , Q ) ), and discover that E 0 ( P , Q ) = lim p → 0 E p ( P , Q ) is merely the curvature entropy E ( P , Q ) defined by Ma et al. [13]. Moreover, by combining the two classes of convex bodies defined by Yang and Zhang [9], we generalize inequality (1.5) to the three-dimensional case.

2 Preliminaries

In Section 2, we present some symbols, relevant basic concepts, and conclusions, for details see [14–17].

Let P ∈ C n , denote by h P the support function of P , then h P : R n → R and

h P ( x ) = max { x ⋅ y ∣ y ∈ P } .

Assuming ∂ ′ P is the set of boundary points of P with a unique unit outer normal, and ℋ n − 1 is the ( n − 1 ) -dimensional Hausdorff measure. If g P : ∂ ′ P → S n − 1 is the Gauss map, and w ⊆ S n − 1 is a Borel set, then the surface area measure S P of P is expressed as

S P ( w ) = ℋ n − 1 ( g P − 1 ( w ) ) .

If P ∈ C o n , then its cone-volume measure V P is expressed as

V P ( w ) = 1 n ∫ y ∈ g P − 1 ( w ) y ⋅ g P ( y ) d ℋ n − 1 ( y ) .

Hence,

d V P = 1 n h P d S P ,

V ( P ) = 1 n ∫ S n − 1 h P ( u ) d S P ( u ) ,

and the normalized cone-volume measure V ¯ P of P is given by

d V ¯ P ( ⋅ ) = 1 V ( P ) d V P ( ⋅ ) .

Moreover, if P has strict convexity and its boundary ∂ P is C 2 , then the Gauss map g : ∂ P → S n − 1 is reversible. Hence, for any x ∈ ∂ P , there is a unique u in S n − 1 that makes x = g − 1 ( u ) . Denote by H n − 1 ( x ) the Gauss curvature of the boundary ∂ P at x , the surface area element d S P ( x ) (or d S P ( g − 1 ( u ) ) ) of P and d u of S n − 1 at u the surface area element are related by

d S P ( x ) = 1 H n − 1 ( g − 1 ( u ) ) d u .

Let h i j be the covariant derivative of h under an orthonormal frame on S n − 1 and δ i j the Kronecker delta function. Then, the reciprocal Gauss curvature can be expressed as [18]

1 H n − 1 = det ( h i j + h P δ i j )

and

d S P ( x ) = det ( h i j + h P δ i j ) d u .

Furthermore, one can obtain

V ( P ) = 1 n ∫ S n − 1 h P det ( h i j + h P δ i j ) d u .

Let P , Q ∈ C o n . The mixed cone-volume measure V P , Q and the first mixed volume V 1 ( P , Q ) are given by [19]

V P , Q ( w ) = 1 n ∫ w h Q ( u ) d S P ( u )

and

V 1 ( P , Q ) = 1 n ∫ S n − 1 h Q ( u ) d S P ( u ) .

Naturally, the normalized cone-volume measure V ¯ P , Q ( w ) can be expressed as

V ¯ P , Q ( w ) = V P , Q ( w ) V 1 ( P , Q ) .

For t ≥ 0 , if P + t Q is an outer parallel convex body of P relative to Q , then its volume can be expressed as an n -degree polynomial of t

V ( P + t Q ) = ∑ i = 0 n n i W i ( P , Q ) t i .

This is exactly the relative Steiner formula, where W i ( P , Q ) are relative quermassintegrals of P to Q for i = 0 , … , n . Particularly, W 0 ( P , Q ) = V ( P ) , W n ( P , Q ) = V ( Q ) , and W i ( P , Q ) = W n − i ( Q , P ) . It is easy to see that V 1 ( P , Q ) = W 1 ( P , Q ) = W n − 1 ( Q , P ) .

When n = 3 , W 1 ( P , Q ) and W 2 ( P , Q ) are expressed as [16]

(2.1) V 1 ( P , Q ) = W 1 ( P , Q ) = 1 3 ∫ S 2 h Q ( u ) d S P ( u ) ,

(2.2) V 1 ( Q , P ) = W 2 ( P , Q ) = 1 3 ∫ S 2 h P ( u ) d S Q ( u ) .

In 2019, Yang and Zhang [9] defined the i th relative Bonnesen function ℬ i ; P , Q ( t ) by

(2.3) ℬ i ; P , Q ( t ) = 2 W i + 1 ( P , Q ) t − W i ( P , Q ) − W i + 2 ( P , Q ) t 2 .

Denote by o the origin, denote by r o ( P , Q ) and R o ( P , Q ) the relative inner radius and the relative outer radius of P to Q , then

r o ( P , Q ) = min u ∈ S n − 1 h P ( u ) h Q ( u )

and

R o ( P , Q ) = max u ∈ S n − 1 h P ( u ) h Q ( u ) .

Definition 2.1

[9] Let P , Q ∈ C o 3 and the origin o ∈ int ( P ∩ Q ) , then P is said to be in R 1 class with respect to Q if

ℬ 0 ; P , Q ( r ) ≥ 0 , r ∈ [ r o ( P , Q ) , R o ( P , Q ) ] ;

P is said to be in R 2 class with respect to Q if

ℬ 1 ; P , Q ( r ) ≥ 0 , r ∈ [ r o ( P , Q ) , R o ( P , Q ) ] .

In [9], the authors obtained the log-Minkowski inequalities for two classes of convex bodies above, see Lemmas 2.2 and 2.3.

Lemma 2.2

Let P , Q ∈ C o 3 . If P is in R 1 with respect to Q , then

(2.4) ∫ S 2 h P 2 h Q d S P ≤ 3 V ( P ) V 1 ( P , Q ) V 1 ( Q , P ) .

If P is in R 2 with respect to Q, then

(2.5) ∫ S 2 h P 2 h Q d S P ≤ 6 V ( P ) V 1 ( Q , P ) − 3 V 1 ( P , Q ) 2 V ( Q ) ,

and equalities in (2.4) and (2.5) hold when P and Q are dilates.

Lemma 2.3

Let P , Q ∈ C o 3 . If P is in R 1 with respect to Q, then

(2.6) ∫ S 2 log h Q h P d V P ≥ V ( P ) 3 log V ( Q ) V ( P )

with equality holds when P and Q are dilates.

3 Mixed curvature entropy of convex bodies

In Section 3, for P , Q ∈ C o n , we define the mixed curvature entropy M f ( P , Q ) and its normalization M ¯ f ( P , Q ) and discuss their fundamental properties when f = x p . Furthermore, we discover that lim p → 0 M ¯ p ( P , Q ) is interesting and establish its log-Minkowski equalities for some three-dimensional convex bodies.

Definition 3.1

Let P , Q ∈ C o n with C 2 boundary and f ( x ) : R + → R be a continuous function. The mixed curvature entropy M f ( P , Q ) of P , Q about f is defined by

(3.1) M f ( P , Q ) = ∫ S n − 1 f H n − 1 ( P ) H n − 1 ( Q ) d V P , Q .

Moreover, the normal mixed curvature entropy M ¯ f ( P , Q ) is defined by

(3.2) M ¯ f ( P , Q ) = 1 V 1 ( P , Q ) M f ( P , Q ) = ∫ S n − 1 f H n − 1 ( P ) H n − 1 ( Q ) d V ¯ P , Q .

By Lemma 2.2, we can obtain the following proposition.

Proposition 3.2

Let P , Q ∈ C o 3 . If P is in R 1 with respect to Q and f ( x ) : R + → R is concave (or convex) and strictly increasing (or decreasing), then

(3.3) ∫ S 2 f h P 2 h Q 2 d V ¯ P , Q ≤ f V ( P ) V 1 ( Q , P ) ,

(3.4) o r ∫ S 2 f h P 2 h Q 2 d V ¯ P , Q ≥ f V ( P ) V 1 ( Q , P ) .

If P is in R 2 with respect to Q and f is concave (or convex) and strictly increasing (or decreasing), then

(3.5) ∫ S 2 f h P 2 h Q 2 d V ¯ P , Q ≤ f 2 V ( P ) V 1 ( Q , P ) − V 1 ( P , Q ) 2 V ( Q ) V 1 ( P , Q ) ,

(3.6) o r ∫ S 2 f h P 2 h Q 2 d V ¯ P , Q ≥ f 2 V ( P ) V 1 ( Q , P ) − V 1 ( P , Q ) 2 V ( Q ) V 1 ( P , Q ) ,

and equalities (3.3)–(3.6) hold when P and Q are dilates.

For examples, if P is in R 1 with respect to Q , in Proposition 3.2 taking f ( x ) = 1 x , then

∫ S 2 h Q 2 h P 2 d V ¯ P , Q ≥ V 1 ( Q , P ) V ( P ) .

Furthermore, taking f ( x ) = log x , then

∫ S 2 log h P 2 h Q 2 d V ¯ P , Q ≤ log V ( P ) V 1 ( Q , P ) ,

and taking f ( x ) = x ,

∫ S 2 h P h Q d V ¯ P , Q ≤ V ( P ) V 1 ( Q , P ) .

For the general bodies P , Q ∈ C o n with C 2 boundary, considering f ( x ) = x p , then the normal mixed curvature entropy and the mixed curvature entropy, M ¯ p ( P , Q ) and M p ( P , Q ) , can be expressed, respectively, as

(3.7) M ¯ p ( P , Q ) = ∫ S n − 1 H n − 1 ( P ) H n − 1 ( Q ) p d V ¯ P , Q 1 p

and

(3.8) M p ( P , Q ) = V 1 ( Q , P ) 1 p M ¯ p ( P , Q ) = ∫ S n − 1 H n − 1 ( P ) H n − 1 ( Q ) p d V P , Q 1 p ,

when 0 < p < q , Jesson’s inequality shows that M ¯ p ( P , Q ) ≤ M ¯ q ( P , Q ) , which together with

∫ S n − 1 H n − 1 ( P ) H n − 1 ( Q ) d V ¯ P , Q = V ( Q ) V 1 ( P , Q )

gives us Proposition 3.3.

Proposition 3.3

Let P , Q ∈ C o n with C 2 boundary. If 0 < p < 1 , then

(3.9) M ¯ p ( P , Q ) ≤ M ¯ 1 ( P , Q ) = V ( Q ) V 1 ( P , Q )

and if p ≥ 1 ,

(3.10) M ¯ p ( P , Q ) ≥ V ( Q ) V 1 ( P , Q ) .

Now, we define the normal mixed log curvature entropy, M ¯ 0 ( P , Q ) , of P and Q by taking the limit on both sides of the above equation (3.7) as p → 0 , that is,

(3.11) M ¯ 0 ( P , Q ) ≕ lim p → 0 log M ¯ p ( P , Q ) = ∫ S n − 1 log H n − 1 ( P ) H n − 1 ( Q ) d V ¯ P , Q .

Moreover, we can obtain the mixed log curvature entropy M 0 ( P , Q ) by

M 0 ( P , Q ) = V 1 ( P , Q ) M ¯ 0 ( P , Q ) .

Theorem 3.4

Let P , Q ∈ C o 3 with C 2 boundary. If P is in R 1 with respect to Q, then

(3.12) M 0 ¯ ( P , Q ) ≤ log V ( Q ) V ( P ) V 1 ( P , Q ) 2 .

If P is in R 2 with respect to Q, then

(3.13) M 0 ¯ ( P , Q ) ≤ log 2 V ( Q ) V 1 ( P , Q ) − V 1 ( Q , P ) 2 V 1 ( P , Q ) V 1 ( Q , P ) .

The two equal signs above hold if P and Q are dilates.

Proof

From the definition of the normal mixed log curvature entropy, we obtain

(3.14) M 0 ¯ ( P , Q ) − ∫ S 2 log h P h Q d V ¯ P , Q = ∫ S 2 log H 2 ( P ) H 2 ( Q ) h Q h P d V ¯ P , Q .

Applying the Jensen inequality, we have

(3.15) ∫ S 2 log H 2 ( P ) H 2 ( Q ) h Q h P d V ¯ P , Q ≤ log ∫ S 2 H 2 ( P ) H 2 ( Q ) h Q h P d V ¯ P , Q = log 1 3 V 1 ( P , Q ) ∫ S 2 H 2 ( P ) h Q 2 H 2 ( Q ) h P d S P = log 1 3 V 1 ( P , Q ) ∫ S 2 h Q 2 H 2 ( Q ) h P d u = log 1 3 V 1 ( P , Q ) ∫ S 2 h Q 2 h P d S Q .

Combining (3.14) with (3.15), we can obtain

M 0 ¯ ( P , Q ) − ∫ S 2 log h P h Q d V ¯ P , Q ≤ log 1 3 V 1 ( P , Q ) ∫ S 2 h Q 2 h P d S Q ,

which together with the fact ∫ S 2 h P h Q d V ¯ P , Q = V ( P ) V 1 ( P , Q ) gives us

(3.16) M 0 ¯ ( P , Q ) − log V ( P ) V 1 ( Q , P ) ≤ log 1 3 V 1 ( P , Q ) ∫ S 2 h Q 2 h P d S Q .

If P is in R 1 with respect to Q , from (2.4), one can obtain

(3.17) ∫ S 2 h Q 2 h P d S Q ≤ 3 V ( Q ) V 1 ( Q , P ) V 1 ( P , Q ) ,

which together with monotonicity of the logarithmic function gives us

M 0 ¯ ( P , Q ) ≤ log V ( Q ) V ( P ) V 1 ( P , Q ) 2 .

So, we complete the proof of (3.12).

If P is in R 2 with respect to Q , by (2.5), one can obtain

(3.18) ∫ S 2 h Q 2 h P d S Q ≤ 6 V ( Q ) V 1 ( P , Q ) − 3 V 1 ( Q , P ) 2 V ( P ) .

Combining (3.16) with (3.18), we can obtain

M 0 ¯ ( P , Q ) − log V ( P ) V 1 ( Q , P ) ≤ log 2 V ( Q ) V 1 ( P , Q ) − V 1 ( Q , P ) 2 V ( P ) V 1 ( P , Q ) ,

that is,

M 0 ¯ ( P , Q ) ≤ log 2 V ( Q ) V 1 ( P , Q ) − V 1 ( Q , P ) 2 V 1 ( P , Q ) V 1 ( Q , P ) ,

which completes the proof of (3.13).□

4 Curvature entropy of convex bodies

In Section 4, we define the general curvature entropy E f ( P , Q ) and discuss some of its fundamental properties when f = x p . We discover that lim p → 0 E p ( P , Q ) is exactly the curvature entropy E ( P , Q ) in [13]. Then, we generalize inequality (1.5) to the three-dimensional case.

Definition 4.1

Let P , Q ∈ C o n with C 2 boundary and f ( x ) : R + → R be a continuous function. The curvature entropy E f ( P , Q ) of P , Q about f is defined by

(4.1) E f ( P , Q ) ≔ ∫ S n − 1 f H n − 1 ( P ) H n − 1 ( Q ) d V P .

Moreover, the normal curvature entropy E ¯ f ( P , Q ) of P , Q is defined by

(4.2) E ¯ f ( P , Q ) ≔ ∫ S n − 1 f H n − 1 ( P ) H n − 1 ( Q ) d V ¯ P .

Specially, we can define E p ( P , Q ) and E ¯ p ( P , Q ) , respectively,

(4.3) E p ( P , Q ) = ∫ S n − 1 H n − 1 ( P ) H n − 1 ( Q ) p d V P 1 p

and

(4.4) E ¯ p ( P , Q ) = ∫ S n − 1 H n − 1 ( P ) H n − 1 ( Q ) p d V ¯ P 1 p .

If the symbol E ¯ 0 ( P , Q ) represents the limit of log E ¯ p ( P , Q ) as p → 0 , then one can obtain

(4.5) E ¯ 0 ( P , Q ) = lim p → 0 log E ¯ p ( P , Q ) = ∫ S n − 1 log H n − 1 ( P ) H n − 1 ( Q ) d V ¯ P .

Moreover,

(4.6) E 0 ( P , Q ) = ∫ S n − 1 log H n − 1 ( P ) H n − 1 ( Q ) d V P = V ( P ) E ¯ 0 ( P , Q ) .

Remark 4.2

Here, E 0 ( P , Q ) is exactly E ( P , Q ) in [13], and

(4.7) E 0 ( P , Q ) ≤ V ( P ) log V 1 ( Q , P ) V ( P ) .

Taking p = 1 in (4.3), then

E 1 ( P , Q ) = ∫ S n − 1 H n − 1 ( P ) H n − 1 ( Q ) d V P = 1 n ∫ S n − 1 H n − 1 ( P ) H n − 1 ( Q ) h P d S P = 1 n ∫ S n − 1 1 H n − 1 ( Q ) h P d u = 1 n ∫ S n − 1 h P d S Q = V 1 ( Q , P ) .

Hence,

E 1 ( P , Q ) = V 1 ( Q , P ) .

Lemma 4.3

Let P , Q ∈ C o 3 with C 2 boundary, then

(4.8) E 0 ( P , Q ) + ∫ S n − 1 log h Q h P d V P ≤ V ( P ) log V ( Q ) V ( P ) .

Proof

E 0 ( P , Q ) + ∫ S n − 1 log h Q h P d V P = ∫ S n − 1 log H n − 1 ( P ) H n − 1 ( Q ) h Q h P d V P = V ( P ) ∫ S n − 1 log H n − 1 ( P ) H n − 1 ( Q ) h Q h P d V ¯ P ≤ V ( P ) log ∫ S n − 1 H n − 1 ( P ) H n − 1 ( Q ) h Q h P d V ¯ P = V ( P ) log ∫ S n − 1 1 n V ( P ) h Q H n − 1 ( Q ) d u = V ( P ) log ∫ S n − 1 1 n V ( P ) h Q d S Q = V ( P ) log V ( Q ) V ( P ) .

Hence, inequality (4.8) holds.□

If P is in R 1 with respect to Q , it follows from Lemma 2.2 that

(4.9) ∫ S 2 log h P h Q d V P ≤ V ( P ) log V 1 ( P , Q ) V 1 ( Q , P ) .

Combining (4.9) and Lemma 4.3, we can obtain Theorem 4.4.

Theorem 4.4

Let P , L ∈ C o 3 with C 2 boundary. If P is in R 1 with respect to Q, then we can obtain

(4.10) E 0 ( P , Q ) ≤ V ( P ) log V ( Q ) V 1 ( P , Q ) V ( P ) V 1 ( Q , P ) .

Remark 4.5

Applying the Alexandrov-Fenchel inequality to the right-hand side of (4.10), we can obtain

V ( P ) log V ( Q ) V 1 ( P , Q ) V ( P ) V 1 ( Q , P ) ≤ V ( P ) log V 1 ( Q , P ) V ( P ) ,

which implies that inequality (4.10) is stronger than (4.7).

By a direct computation, one can obtain

∫ S 2 log h Q h P d V P ≤ V ( P ) log ∫ S 2 h Q h P d V ¯ P = V ( P ) log 1 V ( P ) ∫ S 2 H 2 ( Q ) H 2 ( P ) d V Q = V ( P ) log V 1 ( P , Q ) V ( P ) ,

which together with Lemma 2.3 gives the Alexandrov-Fenchel inequality for class of R 1 : If P is in R 1 with respect to Q , then

V ( Q ) V ( P ) 2 ≤ V 1 ( P , Q ) 3 .

Similarly, if P is in R 2 with respect to Q , from Lemma 2.2 it follows that

(4.11) ∫ S 2 log h P h Q d V P ≤ V ( P ) log 2 V ( P ) V 1 ( Q , P ) − V 1 ( P , Q ) 2 V ( P ) V ( Q ) .

Combining (4.11) and Lemma 4.3, we can obtain Theorem 4.6.

Theorem 4.6

Let P , Q ∈ C o 3 with C 2 boundary. If P is in R 2 with respect to Q, then

(4.12) E 0 ( P , Q ) ≤ V ( P ) log 2 V ( P ) V 1 ( Q , P ) − V 1 ( P , Q ) 2 V ( P ) 2 .

Remark 4.7

Applying the Alexandrov-Fenchel inequality to the right-hand side of (4.12), we can obtain

V ( P ) V ( Q ) V 1 ( P , Q ) V ( P ) V 1 ( Q , P ) ≤ V ( P ) log V 1 ( Q , P ) V ( P ) ,

which implies that inequality (4.12) is stronger than (4.7).

Funding information: This work was supported by the Excellent Young Talents Fund Program of Higher Education Institutions of Anhui Province (No. gxyqZD2020022) and the University Natural Science Research Project of Anhui Province (No. 2022AH040067).
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results, and manuscript preparation.
Conflict of interest: The corresponding author states that there is no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Received: 2024-05-28

Revised: 2024-09-06

Accepted: 2024-09-07

Published Online: 2024-10-18

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2024-0066

Keywords for this article

log-Minkowski inequality; mixed curvature entropy; curvature entropy

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