Journal of Convex Analysis 18 (2011), No. 3, 823--832
Copyright Heldermann Verlag 2011
Convexity, Local Simplicity, and Reduced Boundaries of Sets
David G. Caraballo
Dept. of Mathematics, Georgetown University, St. Mary's Hall, Washington, D.C. 20057-1233, U.S.A.
We establish fundamental connections between the convexity of a set K in Rn, 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 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 $K$ 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.
Keywords: Locally simple, convex, reduced boundary, measure-theoretic boundary, topological boundary, interior, exterior, support, separation, density.
MSC: 52A20, 28A75
[ Fulltext-pdf (123 KB)] for subscribers only.