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On the size of approximately convex sets in normed spaces

Tom 140 / 2000

S. J. Dilworth, Ralph Howard, James W. Roberts Studia Mathematica 140 (2000), 213-241 DOI: 10.4064/sm-140-3-213-241

Streszczenie

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.

Autorzy

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

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