  Let $\Delta_m$ be the standard $m$-dimensional simplex of non-negative $m+1$
tuples that sum to unity and let $S$ be a nonempty subset of $\Delta_m$. A real
valued function $h$ defined on a convex subset of a real vector space is
$S$-almost convex iff for all $(t_0,...,t_m)\in S$ and $x_0,...,x_m\in C$ the
inequality h(t_0 x_0+ ... +t_m x_m)\leq 1+ t_0 h(x_0)+ ... +t_m h(x_m) holds. A
detailed study of the properties of $S$-almost convex functions is made,
including the constriction of the extremal (i.e. pointwise largest bounded)
$S$-almost convex function on simplices that vanishes on the vertices. In the
special case that $S$ is the barycenter of $\Delta_m$ very explicit formulas
are given for the extremal function and its maximum. This is of interest as the
extremal function and its maximum give the best constants in various geometric
and analytic inequalities and theorems.
