Related papers: $L^p$-polarity, Mahler volumes, and the isotropic …

In recent work with Berndtsson and Rubinstein, a notion of $L^p$-polarity was introduced, with classical polarity recovered in the limit $p\to\infty$, and $L^1$-polarity closely related to Bergman kernels of tube domains. A Santal\'o…

$L^p$-polarity and $L^p$-Mahler volumes were recently introduced by Berndtsson, Rubinstein, and the author as a new approach, inspired by complex geometry, to the Mahler, Bourgain, and Blocki conjectures. This paper serves two purposes.…

This paper is dedicated to study the sine version of polar bodies and establish the $L_p$-sine Blaschke-Santal\'{o} inequality for the $L_p$-sine centroid body. The $L_p$-sine centroid body $\Lambda_p K$ for a star body…

This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler…

We confirm, in dimension two, Blocki's conjectures on sharp lower bounds for Bergman kernels of tube domains. To that end, we verify a broader class of $L^p$-Mahler conjectures due to Berndtsson and the authors, where $p=1$ are Blocki's…

In this expository paper we discuss the volume product P(K) of convex bodies K in $R^n$; this is the product of volumes of K and its polar K*. The Blaschke- Santalo inequalities state that always $ P(K) \le P(B_2)$ and $ P(B_1)\le P(K)$ .…

We use symplectic techniques to obtain partial results on Mahler's conjecture about the product of the volume of a convex body and the volume of its polar. We confirm the conjecture for hyperplane sections or projections of $\ell_p$-balls…

We re-confirm, for the case of the unit p-ball of R^n, one of recent conjectures of G.Kuperberg on centrally symmetric convex bodies.This conjecture was very recently confirmrd for this particular case by D.A.Gutierrez using polygamma…

The first goal of this paper is to improve some of the results in \cite{BCPR}. Namely, we establish the $L_p$-Brunn-Minkwoski inequality for intrinsic volumes for origin-symmetric convex bodies that are close to the ball in the $C^2$ sense…

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…

The $L^p$-Brunn-Minkowski theory for $p\geq 1$, proposed by Firey and developed by Lutwak in the 90's, replaces the Minkowski addition of convex sets by its $L^p$ counterpart, in which the support functions are added in $L^p$-norm.…

We introduce vector space norms associated to the Mahler measure by using the L^p norm versions of the Weil height recently introduced by Allcock and Vaaler. In order to do this, we determine orthogonal decompositions of the space of…

We state a conjecture about the volume of symplectically self-polar convex bodies and show that it is equivalent to Mahler's conjecture concerning the volume of a convex body and its Euclidean polar. We also establish lower and upper bounds…

In this note, we show that the strong Viterbo conjecture holds true on any convex toric domain, and that the Viterbo's volume-capacity conjecture holds for the product of a $1$-unconditional convex body $A\subset\mathbb{R}^{n}$ and its…

In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight $r^{p-1}$ to the previously inaccessible regime $p\in (-1,0)$: if $g:\mathbb R^n\to\mathbb R_+$ is an integrable…

The $L_p$ versions of the support function and polar body are introduced by Berndtsson, Mastrantonis and Rubinstein in \cite{Berndtsson-Mastrantonis-Rubinstein-2023} recently. In this paper, we prove that the $L_p$-support function of the…

For $-1<p<1$ we introduce the concept of a polar $p$-centroid body ${\Gamma^*_p K}$ of a star body $K$. We consider the question of whether ${\Gamma^*_p K}\subset {\Gamma^*_p L}$ implies $\mathrm{vol}(L)\le \mathrm{vol}(K).$ Our results…

The "Mahler volume" is, intuitively speaking, a measure of how "round" a centrally symmetric convex body is. In one direction this intuition is given weight by a result of Santalo, who in the 1940s showed that the Mahler volume is…

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the volume product of centrally symmetric…

We prove Nikol'skii type inequalities which for polynomials on the $n$-dimensional torus $\mathbb{T}^n$ relate the $L^p$-with the $L^q$-norm (with respect to the normalized Lebesgue measure and $0 <p <q < \infty$). Among other things we…