A general theory of almost convex functions

Abstract

Let Δ m = { ( t 0 , … , t m ) ∈ R m + 1 : t i ≥ 0 , ∑ i = 0 m t i = 1 } \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 m -dimensional simplex and let ∅ ≠ S ⊂ ⋃ m = 1 ∞ Δ m \varnothing \ne S\subset \bigcup _{m=1}^\infty \Delta _m . Then a function h : C → R h\colon C\to \mathbf {R} with domain a convex set in a real vector space is S S -almost convex iff for all ( t 0 , … , t m ) ∈ S (t_0,\dots , t_m)\in S and x 0 , … , x m ∈ C x_0,\dots , x_m\in C the inequality \[ h ( t 0 x 0 + ⋯ + t m x m ) ≤ 1 + t 0 h ( x 0 ) + ⋯ + t m h ( x m ) h(t_0x_0+\dots +t_mx_m)\le 1+ t_0h(x_0)+\cdots +t_mh(x_m) \] holds. A detailed study of the properties of S S -almost convex functions is made. If S S contains at least one point that is not a vertex, then an extremal S S -almost convex function E S : Δ n → R E_S\colon \Delta _n\to \mathbf {R} is constructed with the properties that it vanishes on the vertices of Δ n \Delta _n and if h : Δ n → R h\colon \Delta _n\to \mathbf {R} is any bounded S S -almost convex function with h ( e k ) ≤ 0 h(e_k)\le 0 on the vertices of Δ n \Delta _n , then h ( x ) ≤ E S ( x ) h(x)\le E_S(x) for all x ∈ Δ n x\in \Delta _n . In the special case S = { ( 1 / ( m + 1 ) , … , 1 / ( m + 1 ) ) } S=\{(1/(m+1),\dotsc , 1/(m+1))\} , the barycenter of Δ m \Delta _m , very explicit formulas are given for E S E_S and κ S ( n ) = sup x ∈ Δ n E S ( x ) \kappa _S(n)=\sup _{x\in \Delta _n}E_S(x) . These are of interest, as E S E_S and κ S ( n ) \kappa _S(n) are extremal in various geometric and analytic inequalities and theorems.