Ben Li
We discuss a topological structure on families of convex functions and then apply it to show the existence of extrimizers for the functional Santaló inequality with respect to polar transform and its reverse.
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Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke-Santalo inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a "pointwise Prekopa-Leindler inequality" and show a monotonicity property of the multimarginal Blaschke-Santalo functional.
On weighted Blaschke--Santalo and strong Brascamp--Lieb inequalities
本文研究函数型Blaschke-Santaló不等式的新推广，探索其在标准高斯测度经典设定外的应用，涉及加权Blaschke–Santalo不等式与强Brascamp–Lieb不等式。
Let $K$ be a convex body and $K^\circ$ its polar body. Call $ϕ(K)=\frac{1}{|K||K^\circ|}\int_K\int_{K^\circ}< x,y>^2 dxdy$. It is conjectured that $ϕ(K)$ is maximum when $K$ is the euclidean ball. In particular this statement implies the Blaschke-Santalo inequality. We verify this conjecture when $K$ is restricted to be a $p$--ball.
Homogeneous maximizers of the Blaschke--Santalo-type functionals
本文研究了Blaschke—Santalo型函数的极值问题，通过将问题简化为同质情况并扩展对称化论证，探讨了其与多边际最优运输问题的关联。
A Blaschke-Santaló inequality for unconditional log-concave measures
本文综述并证明Blaschke-Santaló不等式对所有无条件对数凹测度成立，回应了Cordero-Erausquin关于偶对数凹测度的提问。
The Blaschke-Santaló Inequality is the assertion that the volume product of a centrally symmetric convex body in Euclidean space is maximized by (and only by) ellipsoids. In this paper we give a Fourier analytic proof of this fact.
It is shown that each monotone Minkowski endomorphism of convex bodies gives rise to an isoperimetric inequality which directly implies the classical Urysohn inequality. Among this large family of new inequalities, the only affine invariant one - the Blaschke-Santaló inequality - turns out to be the strongest one. A further extension of these inequalities to merely weakly monotone Minkowski endomorphisms is proven to be impossible. Moreover, for functional analogues of monotone Minkowski endomorphisms, a family of analytic inequalities for log-concave functions is established which generalizes…
We establish new functional versions of the Blaschke-Santaló inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of K. Ball and we give a simple proof of the case of equality. As a corollary, we get some inequalities for $\log$-concave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman, with its equality case.
This paper is dedicated to study the sine version of polar bodies and establish the $L_p$-sine Blaschke-Santaló inequality for the $L_p$-sine centroid body. The $L_p$-sine centroid body $Λ_p K$ for a star body $K\subset\mathbb{R}^n$ is a convex body based on the $L_p$-sine transform, and its associated Blaschke-Santaló inequality provides an upper bound for the volume of $Λ_p^{\circ}K$, the polar body of $Λ_p K$, in terms of the volume of $K$. Thus, this inequality can be viewed as the "sine cousin" of the $L_p$ Blaschke-Santaló inequality established by Lutwak and Zhang. As $p\rightarrow \inf…
Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in R^n. The corresponding inequality to the conjecture is sometimes called the the reverse Blaschke-Santalo inequality. The conjecture is known in dimension two and in several special cases. In the class of unconditional convex bodies, Saint Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, w…
On the conjectured capillary Blaschke-Santaló inequality
本文证明当θ∈(0,π/2)时，猜想的毛细Blaschke-Santaló不等式对任何无条件严格凸毛细超曲面成立，并指出当θ∈(π/2,π)时毛细体积乘积无上界。
Asymmetric Blaschke-Santaló functional inequalities
本文建立了 Blaschke-Santaló 不等式的泛函非对称版本，并由此重新得到几何不等式、取等情形及 Lutwak-Yang-Zhang 的概率型不等式，同时研究了具有独立意义的 $L_p$ 质心泛函类比。
On the stochastic proof of the Blaschke-Santaló inequality
本文通过简化方法，直接推导出函数Santaló不等式，而非其对偶形式，基于对Borel证明的改进。
Generalized Blaschke--Santaló-type inequalities, without symmetry restrictions
本文通过熵对偶给出Nakamura-Tsuji广义Legendre对偶关系的简短证明，消除对称性假设，并应用于Wasserstein重心问题的Talagrand型不等式及多凸体的几何Blaschke--Santaló型不等式。
We explore an interplay between an analysis of diffusion flows such as Ornstein--Uhlenbeck flow and Fokker--Planck flow and inequalities from convex geometry regarding the volume product. More precisely, we introduce new types of hypercontractivity for the Ornstein--Uhlenbeck flow and clarify how these imply the Blaschke--Santaló inequality and the inverse Santaló inequality, also known as Mahler's conjecture. Motivated the link, we establish two types of new hypercontractivity in this paper. The first one is an improvement of Borell's reverse hypercontractivity inequality in terms of Nelson's…
Blaschke--Santaló inequalities with respect to costs
该论文研究内积替换为代价的泛函Blaschke–Santaló不等式，结合卦限不等式与Steiner对称化给出充分条件，推导乘性代价相关的Brascamp–Lieb不等式改进及Fradelizi–Meyer定理的L^p版本，还分析了具体代价的亏缺分解等内容。
Transport-entropy forms of direct and Converseblaschke-Santal{ó} inequalities
本文研究Blaschke-Santaló不等式及其Mahler猜想对应物的输运-熵表述，得到s-凹函数的直接与逆不等式，并证明Mahler猜想等价于球面上加强的对数Sobolev型不等式。
Motivated by the Blaschke-Santaló inequality, we define for a convex body K in ${\bf R}^n$ and for $t \in {\bf R}$ the Santaló-regions S(K,t) of K. We investigate properties of these sets and relate them to a concept of Affine Differential Geometry, the affine surface area of K.
The Blaschke Santaló inequality and the $L_p$ affine isoperimetric inequalities are major inequalities in convex geometry and they have a wide range of applications. Functional versions of the Blaschke Santaló inequality have been established over the years through many contributions. More recently and ongoing, such functional versions have been established for the $L_p$ affine isoperimetric inequalities as well. These functional versions involve notions from information theory, like entropy and divergence. We list stability versions for the geometric inequalities as well as for their function…
Regular ellipsoids and a Blaschke-Santaló-type inequality for projections of non-symmetric convex bodies
本文将 Pisier 关于覆盖数正则递减的定理推广至非对称凸体，通过引入关于 Santaló 点位于原点的凸体投影的 Blaschke-Santaló 型不等式作为关键新方法，并探讨了其在各向同性凸体平均范数中的应用。
Volume product
本文是关于凸体体积乘积与Blaschke-Santalo不等式的综述，梳理了上下界的经典结论、开放问题、相关证明方法，以及函数形式不等式、极函数概念和不等式的强化版本等研究进展。
On a Santaló point for Nakamura-Tsuji's Laplace transform inequality
本文将 Nakamura-Tsuji 的拉普拉斯变换不等式从偶函数推广到非偶函数，通过研究双重拉普拉斯变换中平移的下确界实现中心化，并应用于逆向超压缩性。
On cost-induced Santaló-type inequalities in Polish measure spaces
该研究提出了成本-Santaló不等式框架，通过迁移原理推导新的泛函Santaló不等式，应用于RCD(K,∞)空间、汉明立方体及最优传输领域，拓展了相关理论并得到具体实例。
For a convex body $K$ in $\mathbb{R}^n$, we introduce and study the extremal general affine surface areas, defined by \[ {\rm IS}_φ(K):=\sup_{K^\prime\subset K}{\rm as}_φ(K),\quad {\rm os}_ψ(K):=\inf_{K^\prime\supset K}{\rm as}_ψ(K) \] where ${\rm as}_φ(K)$ and ${\rm as}_ψ(K)$ are the $L_φ$ and $L_ψ$ affine surface area of $K$, respectively. We prove that there exist extremal convex bodies that achieve the supremum and infimum, and that the functionals ${\rm IS}_φ$ and ${\rm os}_ψ$ are continuous. In our main results, we prove Blaschke-Santaló type inequalities and inverse Santaló type inequal…
Duality and Heat flow
本研究揭示凸函数勒让德变换与热流演化的关系及其在泛函布拉施克-桑塔洛不等式的应用，并描述了该不等式的局部极大值点。
Mixed $f$-divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log concave functions and establish some of their properties. Among them are affine invariant vector entropy inequalities, like new Alexandrov-Fenchel type inequalities and an affine isoperimetric inequality for the vector form of the Kullback Leibler divergence for log concave functions. Special cases of $f$-divergences are mixed $L_λ$-affine surface areas for log concave functions. For those, we establish various affine isoperimetric i…
The affine quermassintegrals associated to a convex body in $\mathbb{R}^n$ are affine-invariant analogues of the classical intrinsic volumes from the Brunn-Minkowski theory, and thus constitute a central pillar of affine convex geometry. They were introduced in the 1980's by E. Lutwak, who conjectured that among all convex bodies of a given volume, the $k$-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases $k=1$ and $k=n-1$ correspond to the classical Blaschke-Santaló and Petty projection inequalities, respectively. In this work we confirm Lutwak's c…
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