Let Delta(m) = {(t(0),..., t(m)) is an element of Rm+1 : t(i) >= 0, Sigma(m)(i=0) ti = 1} be the standard m-dimensional simplex and let circle minus not equal S subset of boolean OR(infinity)(m= 1) Delta(m). Then a function h: C -> R with domain a convex set in a real vector space is S-almost convex iff for all (t(0),..., t(m)) is an element of S and x(0),..., x(m). C the inequality h(t(0)x(0) + center dot center dot center dot + t(m)x(m)) <= 1 + t(0)h(x(0)) = center dot center dot center dot t(m)h(x(m)) 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 E-S: Delta(n) -> R is constructed with the properties that it vanishes on the vertices of Delta(n) and if h: Delta(n). R is any bounded S- almost convex function with h( ek) = 0 on the vertices of. n, then h x) <= E-S(x) for all x is an element of(n). In the special case S = {(1/(m+1),..., 1/(m+ 1))}, the barycenter of Delta(m), very explicit formulas are given for E-S and (KS)(n) = sup(x is an element of Delta). E-S( x). These are of interest, as E-S and (KS)(n) are extremal in various geometric and analytic inequalities and theorems.
