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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.

References [Enhancements On Off] (What's this?)

  • [1] H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418-491. MR 0110078 (22:961)
  • [2] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1977. MR 0473443 (57:13109)
  • [3] V. Guillemin and A. Pollack, Differential topology, Prentice-Hall, Englewood Cliffs, N.J., 1974. MR 0348781 (50:1276)
  • [4] S. Krantz and H. Parks, Distance to $ {C^k}$ hypersurfaces, J. Differential Equations 40 (1981), 116-120. MR 614221 (82h:58005)
  • [5] J. Milnor, Morse theory, Princeton Univ. Press, Princeton, N.J., 1963. MR 0163331 (29:634)

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