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On the size of approximately convex sets in normed spaces

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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 $ℋ(A,Co(A))≥log_2n-1$ and $diam(A)≤C√n(ln n)^2$, 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.
Słowa kluczowe
  • 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.
  • Department of Mathematics University of South Carolina Columbia, SC 29208, U.S.A.
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