Kahane-Khinchin’s Inequality for Quasi-Norms

Abstract

Abstract. We extend the recent results of R. Latała and O. Guédon 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 ⊂ R n 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).

On the Dual Form of ‘Low M* Estimate’ in the Quasi-convex Case

“…[24], ch. 2) one can show that M K ≤ (c/p) 1/p M for some absolute positive constant c. Moreover, using ideas of Lata la ( [15], see also [7]) it was shown in [16] that M K is equivalent to the M q,K for q ∈ (−p, ∞), where…”

Remarks on the Central Limit Theorem for Non-convex Bodies

“…These bodies are related to unit balls of p − norms and were studied in relation to local theory of Banach spaces by Gordon and Lewis [11], Gordon and Kalton [10], Litvak, Milman and Tomczak-Jaegermann [17] and others (see [4], [8], [12], [16], [18]).…”

“…Remark 1.3. In [16] Litvak constructed an example of a p-convex body for which the volume distribution is very different from the convex case. Litvak's work studies the large deviations regime for p-convex distributions, while our work is focused on the central limit theorem.…”

The geometry of p-convex intersection bodies

“…Note also that a p 1 -convex body is p 2 -convex for all 0 < p 2 p 1 . There is an extensive literature on p-convex bodies as well as the closely related concept of quasi-convex bodies, see for example [A], [BBP1], [BBP2], [D], [GoK], [GoL], [GuL], [Ka], [KaT], [KaPR], [Li1], [Li2], [Li3], [LMP], [LMS], [M] and others.…”