Regularity of the distance function

Foote, Robert L.

doi:10.1090/S0002-9939-1984-0749908-9

Regularity of the distance function
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by Robert L. Foote

Proc. Amer. Math. Soc. 92 (1984), 153-155

DOI: https://doi.org/10.1090/S0002-9939-1984-0749908-9

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Abstract:

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

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