Dmitry Ryabogin

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}

On bodies with directly congruent projections and sections

is a strict local minimizer for the Mahler volume product

On the p-independence boundedness property of Calderón-Zygmund theory

vol_n(K)vol_n(K^*)

A Lemma of Nakajima and Süss on Convex Bodies

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

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

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.

On counterexamples in questions of unique determination of convex bodies

Let K and L be two convex bodies in R4, such that their projections onto all 3-dimensional subspaces are directly congruent. We prove that if the set of diameters of the bodies satisfies an additional condition and some projections do not have certain π-symmetries, then K and L coincide up to translation and an orthogonal transformation. We also show that an analogous statement holds for sections of star bodies, and prove the n-dimensional versions of these results.

An asymmetric convex body with maximal sections of constant volume

Abstract For 0 ≦ α < 1 we construct examples of even integrable functions Ω on the unit sphere 𝕊 d-1 with mean value zero satisfying such that the L 2-bounded singular integral operator T Ω given by convolution with the distribution p.v. Ω(x/|x|)|x|-d is not bounded on L p (ℝ d ) when . In particular, we construct operators T Ω that are bounded on L p exactly when p = 2.

Harmonic Analysis and Uniqueness Questions in Convex Geometry

Abstract Let K and L be two convex bodies in ℝn such that their projections onto every (n − 1)-dimensional subspace are translates of each other. Then K is a translate of L. We give a very simple analytic proof of this fact.

Fourier Analytic Methods in the Study of Projections and Sections of Convex Bodies

We show that if