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 constant
C > 0 such that for every
n, for every centered convex body
K in R
n 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 or
Cdovr(
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.