On the level sets of a distance function in a Minkowski space

Authors:
Ronald Gariepy and W. D. Pepe

Journal:
Proc. Amer. Math. Soc. **31** (1972), 255-259

MSC:
Primary 52.50; Secondary 53.00

DOI:
https://doi.org/10.1090/S0002-9939-1972-0287442-5

MathSciNet review:
0287442

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Abstract: Given a closed subset of an -dimensional Minkowski space with a strictly convex or differentiable norm, then, for almost every , the -level set (points whose distance from the closed set is ) contains an open subset which is an dimensional Lipschitz manifold and whose complement relative to the level set has dimensional Hausdorff measure zero. In case and the norm is twice differentiable with bounded second derivative, almost every level set is a 1 manifold.

**[F]**Herbert Federer,*Curvature measures*, Trans. Amer. Math. Soc.**93**(1959), 418–491. MR**0110078**, https://doi.org/10.1090/S0002-9947-1959-0110078-1**[Fe]**S. Ferry,*A Sard's theorem for distance functions*, Notices Amer. Math. Soc.**17**(1970), 841. Abstract #70T-G127.**[K]**B. P. Kufarev and N. G. Nikulina,*Lebesgue measure of subsets of Euclidean space as the maximum variation of the distance function on a closed set*, Dokl. Akad. Nauk SSSR**160**(1965), 1004–1006 (Russian). MR**0174700****[R]**R. Tyrrell Rockafellar,*Convex analysis*, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR**0274683****[W]**F. Wesley Wilson Jr.,*Implicit submanifolds*, J. Math. Mech.**18**(1968/1969), 229–236. MR**0229252**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0287442-5

Keywords:
Distance function,
level set,
Hausdorff measure,
Lipschitz manifold

Article copyright:
© Copyright 1972
American Mathematical Society