Regularity of the distance function

Abstract

A coordinate-free proof is given of the fact that the distance function δ \delta for a C k {C^k} submanifold M M of R n {{\mathbf {R}}^n} is C k {C^k} near M M when k ⩾ 2 k \geqslant 2 . The result holds also when k = 1 k = 1 if M M has a neighborhood with the unique nearest point property. The differentiability of δ \delta in the C 1 {C^1} case is seen to follow directly from geometric considerations.