Let X be a normed space. A subset A of X is approximately convex if
d(ta+(1−t)b,A)≤1 for all a,b∈A and t∈[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 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.Comment: 32 pages. See also http://www.math.sc.edu/~howard