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28 March 2024
 
  » arxiv » math.MG/9807107

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Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls
S. J. Dilworth ; Ralph Howard ; James W. Roberts ;
Date 20 Jul 1998
Subject Metric Geometry MSC-class: 26B25 52A27 (primary), 39B72 41A44 51M16 52A21 52A40 (secondary) | math.MG
AffiliationUniversity of South Carolina
AbstractA real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$ f((x+y)/2) le (f(x)+f(y))/2 + 1. $$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function~$E$ vanishing on the vertices of a simplex. A set $A$ in a normed space is an approximately convex set iff for all $a,bin A$ the distance of the midpoint $(a+b)/2$ to $A$ is $le 1$. The bounds on approximately convex functions are used to show that in $R^n$ with the Euclidean norm, for any approximately convex set $A$, any point $z$ of the convex hull of $A$ is at a distance of at most $[log_2(n-1)]+1+(n-1)/2^{[log_2(n-1)]}$ from $A$. Examples are given to show this is the sharp bound. Bounds for general norms on $R^n$ are also given.
Source arXiv, math.MG/9807107
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