On a version of the slicing problem for the surface area of convex bodies

TL;DR

This paper investigates a surface-area-based slicing inequality for convex bodies, providing negative results for fixed dimensions and exploring related parameters, while also identifying a variant with an affirmative answer.

Contribution

It introduces a surface-area slicing inequality, demonstrates its limitations in fixed dimensions, and explores related geometric parameters and variants.

Findings

Negative answer for fixed dimensions

Optimal bounds for volume-surface area parameters

A variant with an affirmative answer

Abstract

We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $\alpha_n$ depending (or not) on the dimension $n$ so that $$S(K)\leq\alpha_n|K|^{\frac{1}{n}}\max_{\xi\in S^{n-1}}S(K\cap\xi^{\perp })$$ where $S$ denotes surface area and $|\cdot |$ denotes volume. For any fixed dimension we provide a negative answer to this question, as well as to a weaker version in which sections are replaced by projections onto hyperplanes. We also study the same problem for sections and projections of lower dimension and for all the quermassintegrals of a convex body. Starting from these questions, we also introduce a number of natural parameters relating volume and surface area, and provide optimal upper and lower bounds for them. Finally, we show that, in contrast to the previous negative results, a variant of the problem which arises naturally from the surface area version of the equivalence of the isomorphic Busemann--Petty problem with the slicing problem has an affirmative answer.