On the average volume of sections of convex bodies

The average section functional as(K) of a star body in Rn is the average volume of its central hyperplane sections: (asleft( k right) = int_{{S^{n - 1}}} {left| {K cap {xi ^ bot }} right|} dsigma 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 R ⁿ and for every 1 ≤ k ≤ n ? 2,

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

. 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 _k ⁿ ), where L _K is the isotropic constant of K and dovr(K, BP _k ⁿ ) is the outer volume ratio distance of K to the class BP _k ⁿ 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.