Isoperimetry of waists and local versus global asymptotic convex geometries

Details Isoperimetry of waists and local versus global asymptotic convex geometries Isoperimetry of waists and local versus global asymptotic convex geometries

2004-04-27

arxiv.org/abs/math/0404500v1

Mathematics

Functional Analysis

Scientific paper

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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