## 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.