On the size of approximately convex sets in normed spaces

S. J. Dilworth; Ralph Howard; James W. Roberts

ArticleOriginal scientific text

Title

On the size of approximately convex sets in normed spaces

Authors ¹, ¹, ²

Affiliations

Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.
Department of Mathematics University of South Carolina Columbia, SC 29208, U.S.A.

Abstract

Let X be a normed space. A set A ⊆ X is approximately convex} if d(ta+(1-t)b,A)≤1 for all a,b ∈ A and t ∈ [0,1]. We prove that every n-dimensional normed space contains approximately convex sets A with

ℋ (A, C o (A)) \geq \log_{2} n - 1

and

d i a m (A) \leq C \sqrt n {(\ln n)}^{2}

, 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))=(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.

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Title

On the size of approximately convex sets in normed spaces

Affiliations

Abstract

Bibliography