Stability of the reverse Blaschke-Santaló inequality for unconditional convex bodies

Kim, Jaegil; Zvavitch, Artem

doi:10.1090/S0002-9939-2014-12334-3

Stability of the reverse Blaschke-Santaló inequality for unconditional convex bodies
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by Jaegil Kim and Artem Zvavitch PDF

Proc. Amer. Math. Soc. 143 (2015), 1705-1717 Request permission

Abstract:

Mahler’s conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in $\mathbb {R}^n$. The inequality corresponding to the conjecture is sometimes called the reverse Blaschke-Santaló inequality. The conjecture is known to be true in $\mathbb {R}^2$ and for several special cases. In the class of unconditional convex bodies, Saint-Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, which is sufficiently close to an unconditional body, satisfies the reverse Blaschke-Santaló inequality.

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