Artem Zvavitch

A remark on the Mahler conjecture: Local minimality of the unit cube

We prove that the unit cube

Projections of convex bodies and the fourier transform

B^n_{\infty}

GAUSSIAN BRUNN-MINKOWSKI INEQUALITIES

is a strict local minimizer for the Mahler volume product

Non-uniqueness of convex bodies with prescribed volumes of sections and projections

vol_n(K)vol_n(K^*)

Bezout Inequality for Mixed Volumes

in the class of origin symmetric convex bodies endowed with the Banach-Mazur distance.

Do Minkowski averages get progressively more convex

The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections oflp-balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of the recent result of Barthe and Naor on extremal projections oflp-balls, and give a Fourier analytic solution to Shephard’s problem, originally solved by Petty and Schneider and asking whether symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function.

An asymmetric convex body with maximal sections of constant volume

A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski inequality is proved, together with precise equality conditions, and shown to be best possible from several points of view. A possible new Gaussian Brunn-Minkowski inequality is proposed, and proved to be true in some special cases. Throughout the study attention is paid to precise equality conditions and conditions on the coecients of dilatation. Interesting links are found to the S-inequality and the (B) conjecture. An example is given to show that convexity is needed in the (B) conjecture.

An Application of Shadow Systems to Mahler’s Conjecture

We show that if

Supremum of a Process in Terms of Trees

d\ge 4

A DISCRETE VERSION OF KOLDOBSKY'S SLICING INEQUALITY

is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.