On the size of approximately convex sets in normed spaces

Abstract

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of $A$. We prove that every $n$-dimensional normed space contains approximately convex sets $A$ with $\mathcal{H}(A,\Co(A))\ge \log_2n-1$ and $\diam(A) \le C\sqrt n(\ln n)^2$, where $\mathcal{H}$ 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 $\mathcal{H}(A,\Co(A))=\diam(A)=D$. Several results pertaining to the Hyers-Ulam stability theorem are also proved.