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

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 $$S(K)\le {\alpha }_n|K{|}^{\frac{1}{n}}\underset{\xi \in S^{n-1}}{\max }S(K\cap {\xi }^{\perp })$$ 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.