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.
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The author is a Clay Research Fellow, and was also supported by NSF grant #DMS-0456590.
Received: November 2005; Accepted: February 2006
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Klartag, B. On convex perturbations with a bounded isotropic constant. GAFA, Geom. funct. anal. 16, 1274–1290 (2006). https://doi.org/10.1007/s00039-006-0588-1
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DOI: https://doi.org/10.1007/s00039-006-0588-1