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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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 $ K$ of $ \mathbb{R}^{n}$ with non-empty interior is convex if and only if it has locally finite perimeter in $ \mathbb{R}^{n}$ and has a supporting hyperplane through each point of its reduced boundary. This refines the standard result that such a set $ K$ 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 $ K$. 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
PII: S 0002-9939(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.