Geometric & Functional Analysis GAFA

, Volume 16, Issue 6, pp 1274–1290

On convex perturbations with a bounded isotropic constant

Article

DOI: 10.1007/s00039-006-0588-1

Cite this article as:
Klartag, B. GAFA, Geom. funct. anal. (2006) 16: 1274. doi:10.1007/s00039-006-0588-1

Abstract.

Let \( K \subset {\user2{\mathbb{R}}}^{n} \) be a convex body and ɛ  > 0. We prove the existence of another convex body \( K' \subset {\user2{\mathbb{R}}}^{n} \), whose Banach–Mazur distance from K is bounded by 1 + ɛ, such that the isotropic constant of K’ is smaller than \( c \mathord{\left/ {\vphantom {c {{\sqrt \varepsilon }}}} \right. \kern-\nulldelimiterspace} {{\sqrt \varepsilon }} \), where c  > 0 is a universal constant. As an application of our result, we present a slight improvement on the best general upper bound for the isotropic constant, due to Bourgain.

Keywords and phrases.

Slicing problem isotropic constant transportation of measure hyperplane conjecture 

AMS Mathematics Subject Classification.

52A20 (52A38, 46B07) 

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA

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