Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

Abstract

A real-valued function f defined on a convex set K is an approximately convex function iff it satisfiesfx+y2⩽f(x)+f(y)2+1. A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function E vanishing on the vertices of a simplex. A set A in a normed space is an approximately convex set iff for all a, b∈A the distance of the midpoint (a+b)/2 to A is ⩽1. The bounds on approximately convex functions are used to show that in Rn with the Euclidean norm, for any approximately convex set A, any point z of the convex hull of A is at a distance of at most [log2(n−1)]+1+(n−1)/2[log2(n−1)] from A. Examples are given to show this is the sharp bound. Bounds for general norms on Rn are also given.