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Shay Sadovsky

Publications and source records attributed to Shay Sadovsky.

8 recordsLinked to original sources
arXiv2025

Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals

This paper establishes two new geometric inequalities in the dual Brunn-Minkowski theory. The first, originally conjectured by Lutwak, is the Brunn-Minkowski inequality for dual quermassintegrals of origin-symmetric convex bodies. The second, generalizing Ball's volume ratio inequality, is a reverse isoperimetric inequality: among all origin-symmetric convex bodies in John's position, the cube maximizes the dual quermassintegrals.

math.MG↗
arXiv2023

Godbersen's conjecture for locally anti-blocking bodies

In this note we give a short proof of Godbersen's conjecture for the class of locally anti-blocking bodies. We show that all equality cases amongst locally anti-blocking bodies are for simplices, further supporting the conjecture. The proof of equality cases introduces a useful calculation of mixed volumes of aligned simplices.

math.MG↗
arXiv2023

Transport-entropy forms of direct and Converseblaschke-Santal{\'o} inequalities

We explore alternative functional or transport-entropy formulations of the Blaschke-Santal{\'o} inequality and of its conjectured counterpart due to Mahler. In particular, we obtain new direct and reverse Blaschke-Santal{\'o} inequalities for s-concave functions. We also obtain new sharp symmetrized transport-entropy inequalities for a large class of spherically invariant probability measures, including the uniform measure on the unit Euclidean sphere and generalized Cauchy and Barenblatt distributions. Finally, we show that the Mahler's conjecture is equivalent to some reinforced log-Sobolev type inequality on the sphere.

math.FA↗
arXiv2021

A Zoo of Dualities

In this note we study order reversing quasi involutions and their properties. These maps are dualities (order reversing involutions) on their image. We prove that any order reversing quasi involution is induced by a cost. Invariant sets of order reversing quasi involutions are of special interest and we provide several results regarding their existence and uniqueness. We determine when an order reversing quasi involution on a sub-class can be extended to the whole space and discuss the uniqueness of such an extension. We also provide several ways for constructing new order reversing quasi involutions from given ones. In particular, we define the dual of an order reversing quasi involution. Finally, throughout the paper we exhibit a "zoo" of illustrative examples. Some of them are classical, some have recently attracted attention of the convexity community and some are new. We study in depth the new example of dual polarity and obtain a Blaschke-Santal\'o type inequality for a corresponding Gaussian volume product. The unified point of view on order reversing quasi involutions presented in this paper gives a deeper understanding of the underlying principles and structures, offering a new and exciting perspective on the topic, exposing many new research directions.

math.MG↗
arXiv2021

Optimal measure transportation with respect to non-traditional costs

We study optimal mass transport problems between two measures with respect to a non-traditional cost function, i.e. a cost $c$ which can attain the value $+\infty$. We define the notion of $c$-compatibility and strong-$c$-compatibility of two measures, and prove that if there is a finite-cost plan between the measures then the measures must be $c$-compatible, and if in addition the two measures are strongly $c$-compatible, then there is an optimal plan concentrated on a $c$-subgradient of a $c$-class function. This function is the so-called potential of the plan. We give two proofs of this theorem, under slightly different assumptions. In the first we utilize the notion of $c$-path-boundedness, showing that strong $c$-compatibility implies a strong connectivity result for a directed graph associated with an optimal map. Strong connectivity of the graph implies that the $c$-cyclic monotonicity of the support set (which follows from classical reasoning) guarantees its $c$-path-boundedness, implying, in turn, the existence of a potential. We also give a constructive proof, in the case when one of the measures is discrete. This approach adopts a new notion of `Hall polytopes', which we introduce and study in depth, to which we apply a version of Brouwer's fixed point theorem to prove the existence of a potential in this case.

math.MG↗
arXiv2020

A Rockafellar-type theorem for non-traditional costs

In this note, we present a unified approach to the problem of existence of a potential for the optimal transport problem with respect to non-traditional cost functions, that is, costs that assume infinite values. We establish a new method that relies on proving solvability of a special (possibly infinite) family of linear inequalities. When the index set of this family is countable, we give a necessary and sufficient condition on the coefficients that assures the existence of a solution, and which, in the setting of transport theory, we call $c$-path-boundedness. In the case of an uncountable index set, one needs an additional assumption for solvability. We propose a sufficient condition in this case. We note that any set admitting a potential must be $c$-path-bounded, and this condition replaces $c$-cyclic monotonicity from the classical theory, i.e. when the cost is real-valued. Our method also gives a new and elementary proof for the classical results of Rockafellar, Rochet and R\"uschendorf.

math.MG↗
arXiv2020

Geometric Inequalities for Anti-Blocking Bodies

We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godberson's conjecture, near-optimal bounds on Mahler volumes, Saint-Raymond-type inequalities on mixed volumes, and reverse Kleitman inequalities for mixed volumes. We apply our results to the combinatorics of posets and prove Sidorenko-type inequalities for linear extensions of pairs of 2-dimensional posets. The results rely on some elegant decompositions of differences of anti-blocking bodies, which turn out to hold for anti-blocking bodies with respect to general polyhedral cones.

math.MG↗