On the average volume of sections of convex bodies

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The average section functional as(K) of a centered convex body in Rⁿ is the average volume of central hyperplane sections of K: equation* as(K)=∫ₛⁿ⁻¹|K∩ ξ⊥|dσ(ξ).equation* We study the question if there exists an absolute constant C>0 such that for every n, for every centered convex body K in Rⁿ and for every 0<k<n, as(K) Cᵏ|K|ᵏ/ⁿE∈ Grₙ₋ₖ as(K∩ E). 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 Lₖ is the isotropic constant of K and d ovr(K,BPₖⁿ) is the outer volume ratio distance from K to the class BPₖⁿ 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 in the case where K is in some of the classical positions as well as the natural lower dimensional analogue of the average section functional.