ON THE AVERAGE VOLUME OF SECTIONS OF CONVEX BODIES

The average section functional as(K) of a star body in R-n is the average volume of its central hyperplane sections: as(K) = integral(n-1)(S)vertical bar K boolean AND xi (perpendicular to)vertical bar d sigma(xi). We study the question whether there exists an absolute constant C > 0 such that for every n, for every centered convex body K in Rn and for every 1 <= k <= n - 2, as(K) <= C-k vertical bar K vertical bar(k/n) max as(K boolean AND E). E is an element of Gr(n-k) 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 CLK or Cd-ovr(K, BPkn), where L-K is the isotropic constant of K and d(ovr)(K, BPkn) is the outer volume ratio distance of K to the class BPkn of generalized k-intersection bodies. We also compare as(K) to the average of as(K boolean AND 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.