A general theory of almost convex functions

Dilworth, S.; Howard, Ralph; Roberts, James

doi:10.1090/S0002-9947-06-04061-X

A general theory of almost convex functions
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by S. J. Dilworth, Ralph Howard and James W. Roberts PDF

Trans. Amer. Math. Soc. 358 (2006), 3413-3445 Request permission

Abstract:

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.

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