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Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

✍ Scribed by S.J. Dilworth; Ralph Howard; James W. Roberts

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 540 KB
Volume: 148
Category: Article
ISSN: 0001-8708
DOI: 10.1006/aima.1999.1836

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✦ Synopsis

A real-valued function f defined on a convex set K is an approximately convex function iff it satisfies

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, b # A the distance of the midpoint (a+b)Â2 to A is 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)Â2 [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.

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