On the average volume of sections of convex bodies

Abstract

The average section functional as(K) of a star body in Rn is the average volume of its central hyperplane sections: $as\left( k \right) = \int_{{S^{n - 1}}} {\left| {K \cap {\xi ^ \bot }} \right|} d\sigma \left( \xi \right)$. We study the question whether there exists an absolute constantC > 0 such that for every n, for every centered convex body K in Rⁿ and for every 1 ≤ k ≤ n − 2,

$$as\left( K \right) \leqslant {C^k}{\left| K \right|^{\frac{k}{n}}}\mathop {\max }\limits_{|E \in G{r_{n - k}}} {\kern 1pt} as\left( {K \cap E} \right)$$

. We observe that the case k = 1 is equivalent to the hyperplane conjecture. We show that this inequality holds true in full generality if one replaces C by CL _K orCd_ovr(K, BP ⁿ _k ), where L _K is the isotropic constant of K and dovr(K, BP ⁿ _k ) is the outer volume ratio distance of K to the class BP ⁿ _k of generalized k-intersection bodies. We also compare as(K) to the average of as(K ∩ E) over all k-codimensional sections of K. We examine separately the dependence of the constants on the dimension when K is in some classical position. Moreover, we study the natural lower dimensional analogue of the average section functional.