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A general theory of almost convex functions


A 2001 Preprint by S. Dilworth, R. Howard, and J. Roberts

  • 2001:04
  • Let $\Delta _ m=\{(t _ 0,\ldots,t _ m)\in{R^{n+1}}:t _ i\geq0,\sum^m _ {i=0}t _ i=1\}$ be the standard m-dimensional simplex. Let $\emptyset\neq{S}\subset\bigcup^\infty _ {m=1}\Delta _ m$, then a function $h:C\to{R}$ with domain a convex set in a real vector space is S-almost convex iff for all $(t _ 0,\ldots,t _ m)\in{S}$ and $x _ 0,\ldots,x _ m\in{C}$ the inequality

    $$h(t _ 0x _ 0+\cdots+t _ mx _ m)\leq1+t _ 0h(x _ 0)+\cdots+t _ mh(x _ m)$$

    holds. A detailed study of the properties of S-almost convex functions is made. It is also shown that if S contains at least one point that is not a vertex, then an extremal S-almost convex function $E _ s:\Delta _ n\to{R}$ is constructed with the properties that it vanishes on the vertices of $\Delta _ m$ and if $h:\Delta _ n\to{R}$ is any bounded S-almost convex function with $h(e _ k)\leq0$ on the vertices of $\Delta _ n$, then $h(x)\leq{E _ S(x)}$ for all $x\in\Delta _ n$. In the special case $S=\{(\frac{1}{(m+1)},\ldots,\frac{1}{(m+1)})\}$ 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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