A Morse-Sard theorem for the distance function on Riemannian manifolds

Abstract

Abstract. We prove that the set of critical values of the distance function from a submanifold of a complete Riemannian manifold is of Lebesgue measure zero. In this way, we extend a result of Itoh and Tanaka.

Sard theorems for Lipschitz functions and applications in optimization

“…By means of this result, we improve a previous result of Rifford [14,16] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R d to R p that can be expressed as finite selections of C k functions (more generally, continuous selections over a compact countable set).…”

“…Should this be a structural assumption in the spirit of Grothendieck (subanalyticity, tameness of the graph of the Lipschitz function), a strong version of Morse-Sard theorem can then be established (local finiteness of the Clarke critical values), see [3]. Without such structural assumptions, a couple of ad-hoc nonsmooth Morse-Sard results can still be found in the literature for particular Lipschitz functions: the distance function to a Riemanian submanifold [14] and the viscosity solutions of Hamiltonians of certain type [16].…”

“…In this work, we improve the aforementioned results of Rifford, by establishing the following nonsmooth Morse-Sard theorem, which has interesting applications in Riemannian [14] and sub-Riemannian geometry [15] and more generally in the Hamilton-Jacobi theory [16].…”

On Viscosity Solutions of Certain Hamilton–Jacobi Equations: Regularity Results and Generalized Sard's Theorems

“…Furthermore, as for Theorem 1, we mention that as soon as a given viscosity solution of (1) must satisfy a Dirichlet-type condition, we can obtain, under additional assumptions on the data, generalized Sard's theorems. For example, we proved such a result for the case of the distance function to a set N in Riemannian geometry in [39] (compare [29]). In fact, this approach is easily extendable to many other situations.…”

“…Therefore, if u satisfies the generalized Sard Theorem, then almost every level set of u is a locally Lipschitz hypersurface in M . Generalized Sard's theorems have been recently used in [29], [39] and [40] to obtain regularity results on the level sets of distance functions in Riemannian and sub-Riemannian geometry. In the present paper, our aim is to show that in small dimension, sometimes under additional assumptions, any viscosity solution of (1) satisfies the generalized Sard Theorem.…”