#### Abstract

The first variation of the total mass of log-concave functions was studied by Colesanti and Fragalà, which includes the mixed volume of convex bodies. Using Colesanti and Fragalà’s first variation formula, we define the geominimal surface area for log-concave functions, and its related affine isoperimetric inequality is also established.

#### 1. Introduction

As we have known, Minkowski addition (the vector addition of convex bodies) is the cornerstone in the classical Brunn-Minkowski theory. Combining with volume, it leads to the Brunn-Minkowski inequality that is one of the most important results in convex geometry. The first variation of volume with respect to Minkowski addition is named the first mixed volume, and its related inequality is the Minkowski inequality. For more history and developments of the Brunn-Minkowski inequality, one may refer to the excellent survey [1]. For instance, the Prékopa-Leindler inequality [28] is known as the functional version of the Brunn-Minkowski inequality. In recent years, finding the functional counterparts of existing geometric results, especially for log-concave functions, has been receiving intensive attentions (see, e.g., [934]).

In 2013, Colesanti and Fragalà [35] introduced the “Minkowski addition” and “scalar multiplication,” (where ), of log-concave functions and as

We remark that a function is log-concave if it has the form , where is convex. The total mass of is defined as

Similar to the case of convex bodies, Colesanti and Fragalà [35] considered the following variational and it is called the first variation of at along . The first variation,, includes themixed volume when it restrictedandto the subclass of log-concave functions (see [35], Proposition 3.12).

Colesanti and Fragalà’s work inspired us a natural way to extend the geominimal surface area for convex bodies to the class of log-concave functions. For convenience, we recall the definition of geominimal surface area. For a convex body containing the origin in its interior, its geominimal surface area, , is defined as (the case , see Petty [36], and , see Lutwak [37]) where is the volume of the unit ball in -dimensional Euclidean space , is the polar body of defined by , denotes the class of convex bodies in that contain the origin in their interiors, and is the mixed volume (for detailed definition, see Section 2). The fundamental inequality for geominimal surface area is the following affine isoperimetric inequality (see, e.g., [37], Theorem 3.12): with equality if and only if is an ellipsoid.

The geominimal surface area, , is an important notation in the Brunn-Minkowski theory, which serves as a bridge connecting affine differential geometry, relative differential geometry, and Minkowski geometry. In the past three decades, the geominimal surface area has developed rapidly (see [25, 3842] for some of the pertinent results).

Since includes the mixed volume, we extend the geominimal surface area to the functional version as follows.

Definition 1. Let be an integral log-concave function and . The geominimal surface area of is defined as where , and is the polar function of .

In Lemma 5, we prove that the above definition includes the geominimal surface area (4) when and restricted to the subclass of log-concave functions.

In order to study the functional geominimal surface area, we need the integral formula of . Hence, we need some notations. We write for the usual inner product of , and denotes the Euclidean normal of . We say that is an admissible perturbation for if there exists a constant such that is convex, where is the Legendre conjugate of . Let denote the set of log-concave functions given by function such that belongs to

Here, and

Colesanti and Fragalà ([35], Theorem 4.5) provided an integral formula for the first variation when and is an admissible perturbation for . For our aims, we consider the following optimization problem:

If the extremum in (9) exists, then it is denoted by .

In Section 3, we prove that for and , if is finite, then there exists a unique log-concave function such that

Similar to the geometric case, the unique log-concave function is called -Petty functions of and denoted by .

Using -Petty functions, we obtain the following analytic inequality with equality conditions involving .

Theorem 2. Suppose and . If has its barycenter at (i.e., ), then with equality if and for and .

#### 2. Background

##### 2.1. Functional Setting

Letif for everyandit satisfies we say is a convex function; let

By the convexity of , is a convex set. We say that is proper if . The Legendre conjugate of is the convex function defined by

Clearly, for all ; there is an equality if and only if and is in the subdifferential of at . Hence, it can be checked that

On the class of convex functions from to , the infimal convolution is defined by and the right scalar multiplication by a nonnegative real number ,

It was proved in [21] (Proposition 2.1) that if are convex functions and , then

The following result will be used later.

Theorem 3 ([43], Theorem 10.9). Let be a relatively open convex set, and let be a sequence of finite convex functions on . Suppose that the real number is bounded for each . It is then possible to select a subsequence of which converges uniformly on closed bounded subsets of to some finite convex function .

The functional Blaschke-Santaló inequality states that let be nonnegative integrable functions on satisfying

If has its barycenter at , which means that , then with equality if and only if there exists a positive definite matrix and such that, a.e. in ,

##### 2.2. The First Variation of the Total Mass of Log-Concave Functions

In this paper, we set

The total mass functional of is defined as

The Gaussian function plays within class the role of the ball in the set of convex bodies, and . For every , we write

From the definition of polar function and Legendre conjugate of function, we note that if , then

The support function of log-concave function is (see [44])

This is a proper generalization, in the sense that .

Let , and let , then which in explicit form reads

The support function of satisfies

In particular,

Let . The first variation of at along is defined as

The existence of the above limit was proved by Colesanti and Fragalà [35], and with . In particular, for every with , then

The functional version of Minkowski first inequality reads as follows (see, e.g., [35], Theorem 5.1): let and assume that . Then, with equality if and only if there exists such that for .

Let denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the Euclidean space . We write for the set of convex bodies that contain the origin in their interiors. Let denote the -dimensional volume of convex body . The volume of the standard unit ball in is denoted by . A convex body is uniquely determined by its support function, which is defined as , where denotes the usual inner product in . The polar body of is defined by .

For real , , and real , the Minkowski-Firey combination is a convex body whose support function is given by

The mixed volume of convex bodies and is defined by

The existence of this limit is showed in [45].

The following result show that includes the mixed volume for convex bodies.

Proposition 4 ([35], Proposition 3.12). Let and . Let, and . Then, there exists a positive constant such that with , and

We setas the subclasses ofgiven by the function such that belongs to

For log-concave function , the Borel measure on is defined by (see [35])

Here, is the -dimensional Hausdorff measure. We need the fact that the barycenter of is the origin; i.e.,

We recall that the log-concave function is an admissible perturbation for log-concave function if is convex.

Colesanti and Fragalà [35] provided an integral representation of the first variation (see, e.g., [35], Theorem 4.5): let and and assume that is an admissible perturbation for . Then, is finite and is given by

#### 3. Functional Geominimal Surface Areas

Analogy to convex bodies, for and , we consider the following optimization problem:

The following result shows that the above optimization problem includes Lutwak’s geominimal surface areas for convex bodies (4) when (up to a constant which is dependent onand). This is one of the reasons why is called the geominimal surface area for log-concave function .

Lemma 5. Let, , and . If, then with for .

Proof. Let , , and . It is not hard to see that

Then, Proposition 4 tells us that with .

From the definitions of geominimal surface area of convex bodies (4) and log-concave functions (44), we have

Since we need the integral representation of the first variation in dealing the problem (44), we focus on for and . Trivially, .

We need the next lemma.

Lemma 6. Let and assume that is an admissible perturbation for . If, then

Proof. Let and . We note that

Since , we have

The following result shows that the functional geominimal surface area is affine invariant.

Lemma 7. Suppose and . If, then

Proof. By (51) and the definition of polar function (26), we have for . Combing with Lemma 6, we have Therefore, we obtain for .

The following lemma was proved by Cordero-Erausquin and Klartag ([46], Lemma 16).

Lemma 8. Let be a finite Borel measure in , and let be the interior of . If and the barycenter of lies at the origin, then there exists a constant with the following property: for any nonnegative, -integrable, convex function ,

The next proposition shows that the infimum in the definition of the -geominimal surface area of log-concave function is a minimum.

Proposition 9. Let and . If is finite, then there exists a unique log-concave function such that

Proof. From the definition of , there exists a sequence such that , with for all , and Let , then

First, we assume that are nonnegative and for all . In this case, from (14), we have and

Letbe the interior of. By Lemma 8 and ((59)), we conclude thatare uniformly upper bound which is dependent only on. According to Theorem 3, there exists a subsequence that converges pointwise in to a convex function . We extend the definition of by setting for and for ,

This limit always exists in , since the function is nondecreasing for following from the convexity of and . Moreover, we have that as for any . Because is equivalent to (here, ), hence, there exits a log-concave function which satisfies the claim.

In the general case, there exist and such that are nonnegative and for all . The convexity of and ensures the finiteness of ; i.e., for some . Similar to the first case, we have where . Lemma 8 deduces that holds for . Moreover,

Therefore, i.e., for any . Then, along the same line of the first case, we conclude that the claim of this proposition holds.

The uniqueness of the minimizing function is demonstrated as follows. Suppose , such that , and i.e.,

Let and . Define , by

Then, from (18) and (70), we have and by the basic inequality for and (18), we have with equality if and only if . Therefore, is the contradiction that would arise if it was the case that .

The unique function whose existence is guaranteed by Proposition 9 will be denoted by , and will be called the -Petty body of log-concave function (or the -Petty function). The polar function of will be denoted by , rather than . For and , the log-concave function is defined by

Lemma 10. If and , then for ,

Proof. From the definition of and Lemma 7,

Lemma 6 deduces

The uniqueness of Proposition 9 ensures that .

By the Blaschke-Santaló inequality, we obtain the following affine isoperimetric inequality for the functional geominimal surface area.

Theorem 11. Let and . If has its barycenter at , then with equality if and for and .

Proof. Taking in (49), together with (33), we have, i.e., By Blaschke-Santaló inequality (20) and the above inequality, we have

This is the desired inequality.

To obtain the equality condition, first assume that . Formula (77) tells us that

This shows that there is equality in (81). From the condition of Blaschke-Santaló inequality, we known that there exists a positive definite matrix and such that, a.e. in ,

Therefore, we obtain the equality condition, namely, and .

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare no conflict of interest.

#### Authors’ Contributions

All authors contributed equally to this work. All authors have read and agreed to the published version of this manuscript. The second author is the corresponding author.

#### Acknowledgments

N. Fang was supported in part by China Postdoctoral Science Foundation (No. 2019M651001). J. Yang was supported in part by NSFC (No. 11971005).