Abstract

We establish fundamental connections between the convexity of a set K in Rn\mathbb{R}^n, its local simplicity, and its reduced boundary in the sense of geometric measure theory. One of the most important results in convex analysis asserts that a closed set with non-empty interior is convex if and only if it has a supporting hyperplane through each topological boundary point. More generally, requiring only non-empty measure-theoretic interior, we prove that a proper closed subset of Rn\mathbb{R}^n is convex if and only if it is locally simple and has a supporting hyperplane at each point of its reduced boundary, so that the convexity information about a closed set KK is essentially encoded in its reduced boundary. We also use techniques of geometric measure theory to refine and generalize other principal theorems about convex sets, standard results on separation and representation which have found significant applications in functional analysis, economics, optimization, control theory, and other areas. Because convexity is closely related to many other topics, our main theorem helps establish connections between reduced boundaries and these other concepts and results as well.

Author information

Contact details are reproduced from the original publication and may be historical.

David G. Caraballo

Dept. of Mathematics, Georgetown University, St. Mary's Hall, Washington, D.C. 20057-1233, U.S.A.

dgc3@georgetown.edu

Keywords

  • Locally simple
  • convex
  • reduced boundary
  • measure-theoretic boundary
  • topological boundary
  • interior
  • exterior
  • support
  • separation
  • density

Mathematics Subject Classification

52A20, 28A75

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Suggested citation

D. G. Caraballo. “Convexity, Local Simplicity, and Reduced Boundaries of Sets.” Journal of Convex Analysis 18 (2011), No. 3, 823–832. https://doi.org/10.68381/jca18051

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.