Convex bodies

The Lp-Blascke addition for a pair of origin symmetric convex bodies is extended to a pair of convex polytopes containing the origin in their interior and Lp Kneser - Suss inequality for polytopes is established.

5p tamynhan5 10-12-2020 7 1   Download

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 . ...

47p noel_noel 17-01-2013 57 10   Download

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 ⊂...

17p tuanloccuoi 04-01-2013 57 5   Download