Geodesic
On the size of approximately convex sets in normed spaces
Studia Mathematica, Tome 140 (2000) no. 3, pp. 213-241

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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.
DOI : 10.4064/sm-140-3-213-241

S. J. Dilworth 1 ; Ralph Howard 1 ; James W. Roberts 1

1 
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S. J. Dilworth; Ralph Howard; James W. Roberts. On the size of approximately convex sets in normed spaces. Studia Mathematica, Tome 140 (2000) no. 3, pp. 213-241. doi: 10.4064/sm-140-3-213-241

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