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A general theory of almost convex functions

Authors:	S. J. Dilworth Ralph Howard James W. Roberts

Affiliation:	Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208 ; Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208 ; Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Abstract:	Let $Delta_m={(t_0,dots, t_m)in mathbf{R}^{m+1}: t_ige 0, sum_{i=0}^mt_i=1}$ be the standard -dimensional simplex and let $varnothingne Ssubset bigcup_{m=1}^inftyDelta_m$ . Then a function with domain a convex set in a real vector space is -*almost convex* iff for all and the inequality holds. A detailed study of the properties of -almost convex functions is made. If contains at least one point that is not a vertex, then an extremal -almost convex function $E_Scolon Delta_nto mathbf{R}$ is constructed with the properties that it vanishes on the vertices of and if $hcolon Delta_nto mathbf{R}$ is any bounded -almost convex function with on the vertices of , then for all . In the special case , the barycenter of , very explicit formulas are given for and $kappa_S(n)=sup_{xinDelta_n}E_S(x)$ . These are of interest, as and are extremal in various geometric and analytic inequalities and theorems.

Keywords:	Convex hulls convex functions approximately convex functions normed spaces Hyers-Ulam Theorem

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