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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Regularity of the distance function


Author: Robert L. Foote
Journal: Proc. Amer. Math. Soc. 92 (1984), 153-155
MSC: Primary 58C07; Secondary 53A07
MathSciNet review: 749908
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0749908-9
PII: S 0002-9939(1984)0749908-9
Article copyright: © Copyright 1984 American Mathematical Society