Isoperimetry of waists and local versus global asymptotic convex geometries



Duke Mathematical Journal
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Isoperimetry of waists and local versus global asymptotic convex geometries

Roman Vershynin

Source: Duke Math. J. Volume 131, Number 1 (2006), 1-16.

Abstract

If two symmetric convex bodies K and L both have nicely bounded sections, then the intersection of random rotations of K and L is also nicely bounded. For L being a subspace, this main result immediately yields the unexpected existence versus prevalence phenomenon: If K has one nicely bounded section, then most sections of K are nicely bounded. 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). Our method relies on the recent isoperimetry of waists due to M. Gromov [G].

Primary Subjects: 46B07
Secondary Subjects: 52A20

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1134666120
Digital Object Identifier: doi:10.1215/S0012-7094-05-13111-8
Mathematical Reviews number (MathSciNet): MR2219235
Zentralblatt MATH identifier: 05031762

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