Bourgainâ€™s discretization theorem asserts that there exists a universal constant C âˆˆ (0,âˆž) with the following property. Let X,Y be Banach spaces with dimX = n. Fix D âˆˆ (1,âˆž) and set Î´ = eâˆ’nCn .â€¦ (More)

It is shown that if (X, â€–Â·â€–X) is a Banach space with Rademacher cotype q then for every integer n there exists an even integer m . n 1 q such that for every f : Zm â†’ X we have n âˆ‘ j=1 Ex [âˆ¥âˆ¥âˆ¥f (x + mâ€¦ (More)

It is shown that if (X, â€– Â· â€–X) is a Banach space with Rademacher type p > 1 then for every n âˆˆ N there exists an even integer m . n2âˆ’1/p log n such that for every f : Zm â†’ X, Ex,Îµ [âˆ¥âˆ¥âˆ¥f (x + m 2 Îµ )â€¦ (More)

Let T be an n Ã— n random matrix, such that each diagonal entry Ti,i is a continuous random variable, independent from all the other entries of T . Then for every n Ã— n matrix A and every t â‰¥ 0 P [ |â€¦ (More)

Given a closed set C in a Banach space (X, â€– Â· â€–), a point x âˆˆ X is said to have a nearest point in C if there exists z âˆˆ C such that dC(x) = â€–x âˆ’ zâ€–, where dC is the distance of x from C. We shortlyâ€¦ (More)

This note contains two types of small ball estimates for random vectors in finite dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability ofâ€¦ (More)

We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only vectorâ€¦ (More)

In this work, we study the volume ratio of the projective tensor products `p âŠ—Ï€ `q âŠ—Ï€ `r with 1 â‰¤ p â‰¤ q â‰¤ r â‰¤ âˆž. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence ofâ€¦ (More)