Proportional concentration phenomena on the sphere

On the isotropic constant of random polytopes

“…A similar question is studied in [2] (where x is uniformly distributed on S n−1 , but the proof and the estimates for x ∈ B n 2 are similar). We will use [2, Theorem 4.3]: assume that 3 ≤ k ≤ n − 3 and set λ = k/n.…”

Error correction via linear programming

“…The asymptotically sharp estimate was computed by S. Artstein [1], but we will be satisfied with a much weaker elementary estimate, see e.g. [ …”

Asymptotic formulas for the diameter of sections of symmetric convex bodies

“…What we really need is the fact that ( ) = arcsin( √ 1 − ). This already follows by a simple argument: in [1], it is observed that n,k ( ) = Prob(Y n sin 2 ), where Y n is a random variable with distribution Beta (1− )n 2 , n 2 . Since…”

“…For every > 0, among all odd continuous functions f : S k−1 → S n−1 , the -extension of the image f (S k−1 ) in S n−1 has minimal measure if f is the identity function. Using an application of this result by Vershynin [18], together with precise concentration estimates of Artstein [1], we are able to prove the following.…”

“…The asymptotic behaviour of n,k ( ) has been determined by Artstein [1] (see also [19]): Let ∈ (0, 1). Then, the following estimates hold as n → ∞.…”