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On the size of approximately convex sets in normed spaces

A 1999 Preprint by S. Dilworth, R. Howard, and J. Roberts

  • 1999:12
  • Let X be a normed space. A set $A\subseteq{X}$ is approximately convex if $d(ta+(1-t)b,A)\leq1$ for all $a,b\in{A}$ and $t\in[0,1]$. We prove that every n-dimensional normed space contains approximately convex sets A with $H(A,\textrm{Co}(A))\geq\log _ 2n-1$ and $\textrm{diam}(A)\leq{C}\sqrt{n}(\ln{n})^2$, 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 $H(A,\textrm{Co}(A))=\textrm{diam}(A)=D$. Several results pertaining to the Hyers-Ulam stability theorem are also proved.

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