m at h . FA ] 1 0 M ay 2 00 4 A GENERAL THEORY OF ALMOST CONVEX FUNCTIONS

Let ∆m = {(t0, . . . , tm) ∈ R : ti ≥ 0, ∑m i=0 ti = 1} be the standard m-dimensional simplex. Let ∅ 6= S ⊂ ⋃ ∞ m=1 ∆m, then a function h : C → R with domain a convex set in a real vector space is S-almost convex iff for all (t0, . . . , tm) ∈ S and x0, . . . , xm ∈ C the inequality h(t0x0 + · · ·+ tmxm) ≤ 1 + t0h(x0) + · · ·+ tmh(xm) holds. A detailed study of the properties of S-almost convex functions is made. If S contains at least one point that is not a vertex, then an extremal S-almost convex function ES : ∆n → R is constructed with the properties that it vanishes on the vertices of ∆n and if h : ∆n → R is any bounded S-almost convex function with h(ek) ≤ 0 on the vertices of ∆n, then h(x) ≤ ES(x) for all x ∈ ∆n. In the special case S = {(1/(m + 1), . . . , 1/(m + 1))}, the barycenter of ∆m, very explicit formulas are given for ES and κS(n) = supx∈∆n ES(x). These are of interest as ES and κS(n) are extremal in various geometric and analytic inequalities and theorems.