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, 52A30, 60B11 show english descriptions Probabilistic methods in Banach space theory [See also 60Bxx]
Variants of convex sets (star-shaped, ($m, n$)-convex, etc.)
Probability theory on linear topological spaces [See also 28C20] 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]

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