Functional Geominimal Surface Area and Its Related Affine Isoperimetric Inequality

As we have known, Minkowski addition (the vector addition of convex bodies) is the cornerstone in the classical BrunnMinkowski 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 BrunnMinkowski inequality, one may refer to the excellent survey [1]. For instance, the Prékopa-Leindler inequality [2–8] is known as the functional version of the Brunn-Minkowski inequality. In recent years, finding the functional counterparts of existing geometric results, especially for logconcave functions, has been receiving intensive attentions (see, e.g., [9–34]). In 2013, Colesanti and Fragalà [35] introduced the “Minkowski addition” and “scalar multiplication,” α · f ⊕ β · g (where α, β > 0), of log-concave functions f and g as


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 [2][3][4][5][6][7][8] is known as the functional version of the Brunn-Minkowski inequality. In recent years, finding the functional counterparts of existing geometric results, especially for logconcave functions, has been receiving intensive attentions (see, e.g., ).
In 2013, Colesanti and Fragalà [35] introduced the "Minkowski addition" and "scalar multiplication," α · f ⊕ β · g (where α, β > 0), of log-concave functions f and g as We remark that a function f : ℝ n ⟶ ½0,∞Þ is logconcave if it has the form f ðxÞ = e −uðxÞ , where u : ℝ n ⟶ ℝ ∪ f+∞g is convex. The total mass of f 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 J at f along g. The first variation,δJð f , gÞ, includes theL p mixed volume when it restrictedf andgto the subclass of log-concave functions (see [35], Proposition 3.12).
Colesanti and Fragalà's work inspired us a natural way to extend the L p geominimal surface area for convex bodies to the class of log-concave functions. For convenience, we recall the definition of L p geominimal surface area. For a convex body K containing the origin in its interior, its L p geominimal surface area, G p ðKÞ, is defined as (the case p = 1, see Petty [36], and p > 1, see Lutwak [37]) where ω n is the volume of the unit ball in n-dimensional Euclidean space ℝ n , Q ∘ is the polar body of Q defined by Q ∘ = fx ∈ ℝ n : hx, yi ≤ 1,∀y ∈ Qg, K n o denotes the class of convex bodies in ℝ n that contain the origin in their interiors, and V p ðK, QÞ is the L p mixed volume (for detailed definition, see Section 2). The fundamental inequality for L p geominimal surface area is the following affine isoperimetric inequality (see, e.g., [37], Theorem 3.12): with equality if and only if K is an ellipsoid. The L p geominimal surface area, G p ðKÞ, is an important notation in the L p Brunn-Minkowski theory, which serves as a bridge connecting affine differential geometry, relative differential geometry, and Minkowski geometry. In the past three decades, the L p geominimal surface area has developed rapidly (see [25,[38][39][40][41][42] for some of the pertinent results).
Since δJð f , gÞ includes the L p mixed volume, we extend the L p geominimal surface area to the functional version as follows.
Definition 1. Let f : ℝ n ⟶ ½0,∞Þ be an integral log-concave function and p > 0. The L p geominimal surface area of f is defined as where c n = ð2πÞ n/2 , and g ∘ ðxÞ = inf y∈ℝ n e −hx,yi /gðyÞ is the polar function of g.
In Lemma 5, we prove that the above definition includes the L p geominimal surface area (4) when p ≥ 1 and restricted f , g to the subclass of log-concave functions.
In order to study the functional geominimal surface area, we need the integral formula of δJð·, · Þ. Hence, we need some notations. We write hx, yi for the usual inner product of x, y ∈ ℝ n , and ∥x∥ denotes the Euclidean normal of x ∈ ℝ n . We say that g = e −v is an admissible perturbation for f = e −u if there exists a constant c > 0 such that u * − cv * is convex, where u * ðyÞ = sup x∈ℝ n fhx, yi − uðxÞg is the Legendre conjugate of u. Let A ′ denote the set of logconcave functions given by function f such that u = −log f belongs to Here, dom ðuÞ = fx ∈ ℝ n : uðxÞ<+∞g and Colesanti and Fragalà ( [35], Theorem 4.5) provided an integral formula for the first variation δJðf , gÞ when f , g ∈ A ′ and g is an admissible perturbation for f . For our aims, we consider the following optimization problem: ð9Þ If the extremum in (9) exists, then it is denoted by c In Section 3, we prove that for Similar to the geometric case, the unique log-concave function f is called p -Petty functions of f and denoted by T p f .
Using p-Petty functions, we obtain the following analytic inequality with equality conditions involving G ð1Þ p ðf Þ.

Background
2.1. Functional Setting. Letu : ℝ n ⟶ ℝ∪f+∞gif for everyx, y ∈ ℝ n andλ ∈ ½0, 1it satisfies we say u is a convex function; let By the convexity of u, dom ðuÞ is a convex set. We say that u is proper if dom ðuÞ ≠ ∅. The Legendre conjugate of u is the convex function defined by Clearly, uðxÞ + u * ðyÞ ≥ hx, yi for all x, y ∈ ℝ n ; there is an equality if and only if x ∈ dom ðuÞ and y is in the subdifferential of u at x. Hence, it can be checked that On the class of convex functions from ℝ n to ℝ ∪ f+∞g, and the right scalar multiplication by a nonnegative real number α > 0, It was proved in [21] (Proposition 2.1) that if u, v : ℝ n ⟶ ℝ ∪ f+∞g are convex functions and α > 0, then The following result will be used later. The functional Blaschke-Santaló inequality states that let f , g be nonnegative integrable functions on ℝ n satisfying If f has its barycenter at 0, which means that with equality if and only if there exists a positive definite matrix A and C > 0 such that, a.e. in ℝ n , 2.2. The First Variation of the Total Mass of Log-Concave Functions. In this paper, we set The total mass functional of f is defined as The Gaussian function plays within class A the role of the ball in the set of convex bodies, and JðγÞ = ð2πÞ n/2 = c n . For every A ∈ GLðnÞ, we write From the definition of polar function and Legendre conjugate of function, we note that if f ∈ A, then The support function of log-concave function f = e −φ is (see [44]) This is a proper generalization, in the sense that h χ K = h K . Let f = e −u , g = e −v , and let α, β > 0, then which in explicit form reads The support function of α · f ⊕ β · g satisfies In particular, Let f , g ∈ A. The first variation of J at f along g is defined as The existence of the above limit was proved by Colesanti and Fragalà [35], and δJð f , gÞ ∈ ½−k,+∞ with k = max finf ð−log gÞ, 0gJðf Þ. In particular, for every f ∈ A with Jð f Þ > 0, then The functional version of Minkowski first inequality reads as follows (see, e.g., [35], Theorem 5.1): let f , g ∈ A and assume that Jðf Þ > 0. Then, with equality if and only if there exists x 0 ∈ ℝ n such that

Journal of Function Spaces
Let K n denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the Euclidean space ℝ n . We write K n o for the set of convex bodies that contain the origin in their interiors. Let VðKÞ denote the n-dimensional volume of convex body K. The volume of the standard unit ball in ℝ n is denoted by ω n = π n/2 /Γððn/2Þ + 1Þ. A convex body K ∈ K n is uniquely determined by its support function, which is defined as h K ðxÞ = max fhx, yi: y ∈ Kg, where h·, · i denotes the usual inner product in ℝ n . The polar body of K is defined by K ∘ = fx ∈ ℝ n : hx, yi ≤ 1,∀y ∈ Kg.
For real p ≥ 1, K, L ∈ K n , and real ε > 0, the Minkowski-Firey L p combination K+ p ε · L is a convex body whose support function is given by The L p mixed volume V p ðK, LÞ of convex bodies K and L is defined by The existence of this limit is showed in [45].
The following result show that δJð f , gÞ includes the L p mixed volume for convex bodies.

Proposition 4 ([35], Proposition 3.12).
Let q ∈ ð1,+∞Þ, p = q/ðq − 1Þ and K, L ∈ K n o . Let u = ðh K ∘ ðxÞ q Þ/q, vðxÞ = ðh L ∘ ðxÞ q Þ/q, and f = e −u , g = e −v . Then, there exists a positive constant c = cðn, qÞ such that with cðn, qÞ = q n/q Γððn + qÞ/qÞ, and We setA ′ as the subclasses ofAgiven by the functionf such that u = log f belongs to For log-concave function f = e −u , the Borel measure μ f on ℝ n is defined by (see [35]) Here, H n is the n-dimensional Hausdorff measure. We need the fact that the barycenter of μ f is the origin; i.e., We recall that the log-concave function g = e −v is an admissible perturbation for log-concave function f = e −u if is convex. Colesanti and Fragalà [35] provided an integral representation of the first variation δJð·, · Þ (see, e.g., [35], Theorem 4.5): let f = e −u and g = e −v ∈ A ′ and assume that g is an admissible perturbation for f . Then, δJð f , gÞ is finite and is given by

Functional L p Geominimal Surface Areas
Analogy to convex bodies, for f ∈ A and p ∈ ℝ, we consider the following optimization problem: The following result shows that the above optimization problem includes Lutwak's L p geominimal surface areas for convex bodies (4) when p > 1 (up to a constant which is dependent onnandp). This is one of the reasons why G p ð f Þ is called the L p geominimal surface area for log-concave function f .

Journal of Function Spaces
From the definitions of L p geominimal surface area of convex bodies (4) and log-concave functions (44), we have Since we need the integral representation of the first variation δJð f , gÞ in dealing the problem (44), we focus on for f ∈ A ′ and p ∈ ℝ. Trivially, G p ð f Þ ≤ G ð1Þ p ð f Þ. We need the next lemma.
Lemma 6. Let f , g ∈ A ′ and assume that g is an admissible perturbation for f . If A ∈ SLðnÞ, then Proof. Let f ðxÞ = e −uðxÞ and gðxÞ = e −vðxÞ . We note that Since ∇ x ðu ∘ AÞ = A t ∇ Ax u, we have The following result shows that the functional geominimal surface area is affine invariant.
Lemma 7. Suppose f ∈ A ′ and p > 0 . If A ∈ SLðnÞ, then Proof. By (51) and the definition of polar function (26), we have for A ∈ SLðnÞ. Combing with Lemma 6, we have Therefore, we obtain for A ∈ SLðnÞ.

Lemma 8.
Let μ be a finite Borel measure in ℝ n , and let K be the interior of convðSuppðμÞÞ. If x 0 ∈ K and the barycenter of μ lies at the origin, then there exists a constant C μ,x 0 > 0 with the following property: for any nonnegative, μ -integrable, convex function φ : ℝ n ⟶ ℝ ∪ f+∞g, The next proposition shows that the infimum in the definition of the p-geominimal surface area of log-concave function is a minimum. Proposition 9. Let p > 0 and f ∈ A ′ . If Jðf Þ is finite, then there exists a unique log-concave function f ∈ A such that Proof. From the definition of G ð1Þ p ð f Þ, there exists a sequence g i ∈ A ′ such that Jðg ∘ i Þ = c n , with δJð f , γÞ ≥ δJðf , g i Þ for all i, and ð59Þ First, we assume that v i are nonnegative and v i ð0Þ = 0 for all i. In this case, from (14), we have and v * i 0 ð Þ = sup LetKbe the interior ofconvðSuppðμ f ÞÞ. By Lemma 8 and ((59)), we conclude thatv * i are uniformly upper bound which This limit always exists in ½0, +∞, since the function λ ↦ v * ðλxÞ is nondecreasing for λ ∈ ð0, 1Þ following from the convexity of v * and v * ð0Þ = 0. Moreover, we have that v * ðλ xÞZv * ðxÞ as λ ⟶ 1 − for any where Therefore, i.e., for any x 0 ∈ K. 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 h 1 , h 2 ∈ A, such that Jðh ∘ 1 Þ = Jð h ∘ 2 Þ = c n , and Then, from (18) and (70), we have