Reduced boundaries and convexity

Author:
David G. Caraballo

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1775-1782

MSC (2010):
Primary 52A20, 52A30, 28A75

Published electronically:
January 29, 2013

MathSciNet review:
3020862

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Abstract: We establish strong, new connections between convex sets and geometric measure theory. We use geometric measure theory to improve several standard theorems from the theory of convex sets, which have found wide application in fields such as functional analysis, economics, optimization, and control theory. For example, we prove that a closed subset of with non-empty interior is convex if and only if it has locally finite perimeter in and has a supporting hyperplane through each point of its reduced boundary. This refines the standard result that such a set is convex if and only if it has a supporting hyperplane through each point of its topological boundary, which may be much larger than the reduced boundary. Thus, the reduced boundary from geometric measure theory contains all the convexity information for such a set . We similarly refine a standard separation theorem, as well as a representation theorem for convex sets. We then extend all of our results to other notions of boundary from the literature and deduce the corresponding classical results from convex analysis as special cases.

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Additional Information

**David G. Caraballo**

Affiliation:
Department of Mathematics and Statistics, St. Mary’s Hall, 3rd floor, Georgetown University, Washington, DC 20057-1233

DOI:
http://dx.doi.org/10.1090/S0002-9939-2013-11099-3

Keywords:
Convex,
reduced boundary,
measure-theoretic boundary,
topological boundary,
support,
separation,
nearest point,
supporting hyperplane,
supporting half-space,
density

Received by editor(s):
October 26, 2010

Received by editor(s) in revised form:
February 11, 2011

Published electronically:
January 29, 2013

Communicated by:
Tatiano Toro

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.