A general theory of almost convex functions

Let $\Delta_m=\{(t_0,\dots, t_m)\in \mathbf{R}^{m+1}: t_i\ge 0, \sum_{i=0}^mt_i=1\}$ be the standard $m$-dimensional simplex and let $\varnothing\ne S\subset \bigcup_{m=1}^\infty\Delta_m$. Then a function $h\colon C\to \mathbf{R}$ with domain a convex set in a real vector space is $S$-almost convex iff for all $(t_0,\dots, t_m)\in S$ and $x_0,\dots, x_m\in C$ the inequality

$h(t_0x_0+\dots+t_mx_m)\le 1+ t_0h(x_0)+\cdots+t_mh(x_m)$

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 $E_S\colon \Delta_n\to \mathbf{R}$ is constructed with the properties that it vanishes on the vertices of $\Delta_n$ and if $h\colon \Delta_n\to \mathbf{R}$ is any bounded $S$-almost convex function with $h(e_k)\le 0$ on the vertices of $\Delta_n$, then $h(x)\le E_S(x)$ for all $x\in \Delta_n$. In the special case $S=\{(1/(m+1),\dotsc, 1/(m+1))\}$, the barycenter of $\Delta_m$, very explicit formulas are given for $E_S$ and $\kappa_S(n)=\sup_{x\in\Delta_n}E_S(x)$. These are of interest, as $E_S$ and $\kappa_S(n)$ are extremal in various geometric and analytic inequalities and theorems.