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. The result is new even in the standard Euclidean space and optimal in view of regularity.

Reference: T. Miura, M. Tanaka, Delta-convex structure of the singular set of distance functions, Comm. Pure Appl. Math. 77 (2024), no. 9, 3631--3669