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Isoperimetry of waists and local versus global asymptotic convex geometries

Roman Vershynin
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Abstract

Existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of random rotations of K and L is nicely bounded. For L = subspace, this main result immediately yields the unexpected phenomenon: "If K has one nicely bounded section, then most sections of K are nicely bounded". This 'existence implies randomness' consequence was proved independently in [Giannopoulos, Milman and Tsolomitis]. The main result represents a new connection between the local asymptotic convex geometry (study of sections of convex bodies) and the global asymptotic convex geometry (study of convex bodies as a whole). The method relies on the new 'isoperimetry of waists' on the sphere due to Gromov.

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Roman Vershynin. 2004-04-27. Isoperimetry of waists and local versus global asymptotic convex geometries. https://arxiv.org/abs/math/0404500

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