Über die Approximation von lokal konvexen Mengen

Abstract

While convex sets in Euclidean space can easily be approximated by convex sets with C^∞ -boundary, the C^∞ -approximation of convex sets in Riemannian manifolds is a non-trivial problem. Here we prove that C^∞-approximation is possible for a compact, locally convex set C in a Riemannian manifold if (i) C has strictly convex boundary or if (ii) the sectional curvature is positive or negative on C.

The proofs are based on a detailed analysis of the distance function from ∂C, on results from [1] and on the Greene-Wu approximation process for convex functions ([5], [6]). Finally, using similar methods, a partial tubular neighborhood with geodesic fibres is constructed for the boundary of a locally convex set. This construction is essential for some results in [2].