Convex bodies

Xem 1-3 trên 3 kết quả Convex bodies
  • We find a sharp combinatorial bound for the metric entropy of sets in Rn and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central Limit Theorem if the square root of its combinatorial dimension is integrable. 2. The uniform entropy is equivalent to the combinatorial dimension under minimal regularity. Our method also constructs a nicely bounded coordinate section of a symmetric convex body in Rn . ...

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  • Annals of Mathematics By S. Artstein, V. Milman, and S. J. Szarek For two convex bodies K and T in Rn , the covering number of K by T , denoted N (K, T ), is defined as the minimal number of translates of T needed to cover K. Let us denote by K ◦ the polar body of K and by D the euclidean unit ball in Rn . We prove that the two functions of t, N (K, tD) and N (D, tK ◦ ), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K ⊂...

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  • There has been much previous work on node localization and event detection [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] , includ- ing foundational work on theoretical lower bounds [17, 18]. Sex- tant differentiates itself from this body of work in several ways. First, it does not assume uniform transmission radii (i.e.

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