We review the celebrated Johnson Lindenstrauss Lemma and some recent advances in the understanding of probability measures with geometric characteristics on R, for large d. These advances include the central limit theorem for convex sets, according to which the uniform measure on a high dimensional convex body1 has marginals that are approximately Gaussian. We try to combine these two results t...

We extend the recent results of R. Lata la and O. Guédon about equivalence of Lq-norms of logconcave random variables (KahaneKhinchin’s inequality) to the quasi-convex case. We construct examples of quasi-convex bodies Kn ⊂ IRn which demonstrate that this equivalence fails for uniformly distributed vector on Kn (recall that the uniformly distributed vector on a convex body is logconcave). Our e...

The convex body maximal operator is a natural generalization of the Hardy–Littlewood operator. In this paper we are considering its dyadic version in presence matrix weight. To our surprise it turns out that not bounded. This sharp contrast to Doob's inequality context. At first, show Carleson Embedding Theorem with weight fails. We then deduce unboundedness matrix-weighted

The largest volume ratio of a given convex body $K \subset \mathbb R^n$ is defined as $$ \mathrm{lvr}(K):= \mathrm {sup}\_{L R^n} {vr}(K,L), where the sup runs over all bodies $L$. We prove following sharp lower bound: c \sqrt{n} \leq \mathrm{lvr}(K), for every $K$ (where $c > 0$ an absolute constant). This result improves former best known bound, order $\sqrt{{n}/{\log \log(n)}}$. also study e...

In this paper, we give an overview of some results concerning best and random approximation convex bodies by polytopes. We explain how both are linked see that is almost as good approximation.

We first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a Banach space and next show that if the Banach space is having the Opial condition, then the fixed points set of such a mapping with the convex range is nonempty. In particular, we establish that if the Banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...