On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body
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We discuss the analogue in the Brunn-Minkowski theory of the inequalities of Marcus-Lopes and Bergstrom about symmetric functions of positive reals and determinants of symmetric positive matrices respectively. We obtain a local version of the Aleksandrov-Fenchel inequality Wi2 ≥ Wi-1Wi+1 which relates the quermassintegrals of a convex body K to those of an arbitrary hyperplane projection of K. A consequence is the following fact: for any convex body K, for any (n - 1)-dimensional subspace E of ℝn and any t > 0, |PE(K) + tDE|/|PE(K)| ≤ |K + 2tDn|/|K| where D denotes the Euclidean unit ball and |·| denotes volume in the appropriate dimension.
author list (cited authors)
Giannopoulos, A., Hartzoulaki, M., & Paouris, G.