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- W2167809706 abstract "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 $N,{geq}, C(varepsilon )nlog n$ , then [left| I-frac{1}{NL_K^2}sum_{i=1}^Nx_iotimes x_iright| 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_iright|_K,dx_sldots 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." @default.
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- W2167809706 date "2005-12-01" @default.
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- W2167809706 title "RANDOM POINTS IN ISOTROPIC UNCONDITIONAL CONVEX BODIES" @default.
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- W2167809706 doi "https://doi.org/10.1112/s0024610705006897" @default.
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