RANDOM POINTS IN ISOTROPIC UNCONDITIONAL CONVEX BODIES

Autor: M. Hartzoulaki, Antonis Tsolomitis, Apostolos Giannopoulos
Rok vydání: 2005
Předmět:
Zdroj: Journal of the London Mathematical Society. 72:779-798
ISSN: 1469-7750
0024-6107
DOI: 10.1112/s0024610705006897
Popis: The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies $K, T_1,\ldots, T_s$ . (a) Let $\varepsilon\,{\in}\, (0,1)$ and let $x_1,\ldots, x_N$ be chosen from K . Is it true that if $N\,{\geq}\, C(\varepsilon )n\log n$ , then \[\left\| I-\frac{1}{NL_K^2}\sum_{i=1}^Nx_i\otimes x_i\right\| with probability greater than $1\,{-}\,\varepsilon $ ? (b) Let $x_i$ be chosen from $T_i$ . Is it true that the unconditional norm \[\|{\bf t}\|=\int_{T_1}\!{\ldots}\int_{T_s}\left\|\sum_{i=1}^st_ix_i\right\|_K\,dx_s\ldots dx_1\] is well comparable to the Euclidean norm in ${\mathbb R}^s$ ? (c) Let $x_1,\ldots, x_N$ be chosen from K . Let ${\mathbb E}\,(K,N):={\mathbb E}\,|{\rm conv}\{ x_1,\ldots, x_N\}|^{1/n}$ be the expected volume radius of their convex hull. Is it true that ${\mathbb E}\,(K,N)\,{\simeq}\, {\mathbb E}\,(B(n),N)$ for all N , where $B(n)$ is the Euclidean ball of volume 1? It is proved that the answers to these questions are affirmative if there is a restriction to the class of unconditional convex bodies. The main tools come from recent work of Bobkov and Nazarov. Some observations about the general case are also included.
Databáze: OpenAIRE
Externí odkaz: