We discuss the connection between the expansion of small sets in graphs, and the Schatten norms of their adjacency matrices. In conjunction with a variant of the Azuma inequality for uniformly smooth normed spaces, we deduce improved bounds on the small-set isoperimetry of Abelian Alon–Roichman random Cayley graphs.

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 ${{K}_{n\,}}\subset {{\mathbb{R}}^{n}}$ 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).

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.