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Israel Journal of Mathematics

, Volume 222, Issue 2, pp 921–947 | Cite as

On the average volume of sections of convex bodies

  • Silouanos BrazitikosEmail author
  • Susanna Dann
  • Apostolos Giannopoulos
  • Alexander Koldbosky
Article

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 R n and for every 1 ≤ kn − 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 orCdovr(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(KE) 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.

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Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  • Silouanos Brazitikos
    • 1
    Email author
  • Susanna Dann
    • 2
  • Apostolos Giannopoulos
    • 1
  • Alexander Koldbosky
    • 3
  1. 1.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece
  2. 2.Institute of Discrete Mathematics and GeometryVienna University of TechnologyViennaAustria
  3. 3.Department of MathematicsUniversity of MissouriColumbiaUSA

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