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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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Keywords: Distance function, level set, Hausdorff measure, Lipschitz manifold
Article copyright: © Copyright 1972 American Mathematical Society

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