Convexity, Local Simplicity, and Reduced Boundaries of Sets

D. G. Caraballo

Journal of convex analysis, Tome 18 (2011) no. 3, pp. 823-832

Résumé

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.

Détail

Classification : 52A20, 28A75
Mots-clés : Locally simple, convex, reduced boundary, measure-theoretic boundary, topological boundary, interior, exterior, support, separation, density