Delta-convex structure of the singular set of distance functions

2024/05/31 Fri 16:00 - 17:00
Tatsuya Miura
Kyoto University

This talk is about the structure of the singular set of the distance function from an arbitrary closed subset of the standard Euclidean space, or more generally of a complete Finsler manifold. In terms of PDE, the distance function can be viewed as a viscosity solution to the classical eikonal equation or its generalization. Our main result, obtained jointly with Minoru Tanaka (Tokai University), shows that the singular set is equal to a countable union of delta-convex hypersurfaces up to an exceptional set of codimension two. A finer structure theorem is given in two dimensions. Those results are new even in the standard Euclidean space and optimal in view of regularity.