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# Kahane-Khinchin's Inequality for Quasi-Norms

Published:2000-09-01
Printed: Sep 2000
• A. E. Litvak
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## Abstract

We extend the recent results of R.~Lata{\l}a and O.~Gu\'edon about equivalence of $L_q$-norms of logconcave random variables (Kahane-Khinchin's inequality) to the quasi-convex case. We construct examples of quasi-convex bodies $K_n \subset \R$ which demonstrate that this equivalence fails for uniformly distributed vector on $K_n$ (recall that the uniformly distributed vector on a convex body is logconcave). Our examples also show the lack of the exponential decay of the tail" volume (for convex bodies such decay was proved by M.~Gromov and V.~Milman).
 MSC Classifications: 46B09 - Probabilistic methods in Banach space theory [See also 60Bxx] 52A30 - Variants of convex sets (star-shaped, ($m, n$)-convex, etc.) 60B11 - Probability theory on linear topological spaces [See also 28C20]