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 \begin{equation*} S(K)\leqslant \alpha _n|K|^{\frac {1}{n}}\max _{\xi \in S^{n-1}}S(K\cap \xi ^{\perp }), \end{equation*} 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.