A General Theory of Almost Convex Functions

Let ∆m = {(t0, . . . , tm) ∈ Rm+1 : ti ≥ 0, ∑m i=0 ti = 1} be the standard m-dimensional simplex and let ∅ = 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.