Reduced boundaries and convexity

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.