Estimating volume and surface area of a convex body via its projections or sections

TL;DR

This paper establishes new inequalities relating the surface area of convex bodies to their projections and sections, addresses an open question on the reverse Loomis-Whitney inequality, and explores conditions for the slicing problem.

Contribution

It introduces novel inequalities connecting surface measures of convex bodies with their projections and sections, and provides new criteria for the slicing problem based on outer volume ratio distances.

Findings

Solved an open question on the asymptotic behavior of the reverse Loomis-Whitney inequality.

Provided a new sufficient condition for the slicing problem involving intersection bodies.

Demonstrated that volume ratio and minimal surface area are not necessarily close.

Abstract

The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P. Gritzmann and P. Gronchi regarding the asymptotic behavior of the best constant in a recently proposed reverse Loomis-Whitney inequality. Next we give a new sufficient condition for the slicing problem to have an affirmative answer, in terms of the least "outer volume ratio distance" from the class of intersection bodies of projections of at least proportional dimension of convex bodies. Finally, we show that certain geometric quantities such as the volume ratio and minimal surface area (after a suitable normalization) are not necessarily close to each other.