Size of Approximately Convex Sets in Normed Spaces (Preprint).

摘要

Let X be a normed space. A set A subset of X is approximately convex if d(ta + (1 - t)b,A) is less or equal to 1 for all a, b elements of A and t element of [0, 1]. We prove that every n-dimensional normed space contains approximately convex sets A with H(A, Co(A)) greater or equal to log2 n - 1 and diam(A) smaller or equal to C(square root of n)(ln n)squared where H denotes the Hausdorff distance. These estimates are reasonably sharp. For every D >0, we construct worst possible approximately convex sets in C(0, 1) such that H(A, Co(A)) = diam(A) = D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.