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

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
MathSciNet review: 0287442
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Abstract: Given a closed subset of an $ n$-dimensional Minkowski space with a strictly convex or differentiable norm, then, for almost every $ r > 0$, the $ r$-level set (points whose distance from the closed set is $ r$) contains an open subset which is an $ n - 1$ dimensional Lipschitz manifold and whose complement relative to the level set has $ n - 1$ dimensional Hausdorff measure zero. In case $ n = 2$ and the norm is twice differentiable with bounded second derivative, almost every level set is a 1 manifold.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0287442-5
PII: S 0002-9939(1972)0287442-5
Keywords: Distance function, level set, Hausdorff measure, Lipschitz manifold
Article copyright: © Copyright 1972 American Mathematical Society