We study functional Blaschke--Santaló inequalities in which the inner product is replaced by a cost. An inequality on each orthant, combined with Steiner symmetrization, gives a sufficient condition for a cost inequality to hold for even convex functions. We apply this principle to a multiplicative cost defined on all orthants. Its second variation gives a weighted improvement of the Brascamp--Lieb inequality in the directions for which differentiation is justified. We also prove an $L^p$ version of the Fradelizi--Meyer theorem from a weighted volume-product inequality and derive the corresponding radial cost and Brascamp--Lieb inequalities. Explicit examples include anisotropic power and Cauchy densities, with dilation equality cases and a sharp obstruction below the coordinate exponent two. For the spherical logarithmic cost we compute an exact entropy--logdet deficit decomposition and the second-order deficit transverse to ellipsoids. We also explain the transport-entropy formulation and distinguish the restricted spectral question from the scalar $(B)$ property.
Stability versions of the functional forms of the Blaschke-Santalo inequality due to Ball, Artstein-Klartag-Milman, Fradelizi-Meyer and Lehec are proved.
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…
We present a complete 3-dimensional Blaschke-Santaló diagram for planar convex bodies with respect to the four classical magnitudes inner and outer radius, diameter and (minimal) width in euclidean spaces.
We prove the Blaschke-Santaló inequality restricted to $n$-gons: the extremal polygons are the affine regular $n$-gons. If either the John or the Löwner ellipse of a planar $o$-symmetric convex body $K$ is the unit circle about $o$, then a sharpening of the Blaschke-Santaló inequality holds: even the aritmetic mean $\left( V(K) + V( K^*) \right) /2 $ is at least $π$. We give stability variants of the Blaschke-Santaló inequality for the plane. If for some $n \ge 3$ the planar convex body $K$ is $n$-fold rotationally symmetric about $o$, then we give the exact maximum of $V(K^*)$, as a function …
A stability version of the Blaschke-Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n>2 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.
We provide a study of Blaschke-Santaló diagrams for the inradius, diameter, and circumradius, measured with respect to different gauges. This contrasts previous works on those diagrams, which are all considered for euclidean measure. By proving several new inequalities and properties between these three functionals, we compute the intersection and the union over all possible gauges of those diagrams, showing that they coincide with the corresponding diagrams of a parallelotope and (in the planar case) a triangle, respectively. Further new extremal properties are derived considering the diagram…
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.
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…
On cost-induced Santaló-type inequalities in Polish measure spaces
Carnegie Mellon University（卡内基梅隆大学）；University of Toulouse & CNRS（图卢兹大学及法国国家科学研究中心）；University of Cincinnati（辛辛那提大学）；The Ohio State University（俄亥俄州立大学）
Let $k\geq 2$ be an integer. In the spirit of Kolesnikov-Werner \cite{KW}, for each $j\in\{2,\ldots,k\}$, we conjecture a sharp Santaló type inequality (we call it $j$-Santaló conjecture) for many sets (or more generally for many functions), which we are able to confirm in some cases, including the case $j=k$ and the unconditional case. Interestingly, the extremals of this family of inequalities are tuples of the $l_j^n$-ball. Our results also strengthen one of the main results in \cite{KW}, which corresponds to the case $j=2$. All members of the family of our conjectured inequalities can be i…
H. Guggenheimer generalized the planar volume product problem for locally convex curves $C$ enclosing the origin $k \ge 2$ times. He conjectured that the minimal volume product $V(C)V(C^*)$ for these curves is attained if the curve consists of the longest diagonals of a regular $(2k+1)$-gon, with centre $0$, these diagonals taken always in the positive orientation. This conjectured minimum is of the form $k^2 + O(k)$. We investigate special cases of this conjecture. We prove it for locally convex $n$-gons with $2k+1 \le n \le 4k$, if the central angles at $0$ of all sides are equal to $2k π/n$…
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…
In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ affine surface area -- a fundamental object in (affine) convex geometry. We prove some properties for the $L_p$ geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and a Santaló style inequality. Cyclic inequalities are established to…
