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

Geodesics in Euclidean space with analytic obstacle


Authors: Felix Albrecht and I. D. Berg
Journal: Proc. Amer. Math. Soc. 113 (1991), 201-207
MSC: Primary 53C22
MathSciNet review: 1077783
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Abstract: In this note we are concerned with the behavior of geodesies in Euclidean $ n$-space with a smooth obstacle. Our principal result is that if the obstacle is locally analytic, that is, locally of the form $ {x_n} = f({x_1}, \ldots ,{x_{n - 1}})$ for a real analytic function $ f$, then a geodesic can have, in any segment of finite arc length, only a finite number of distinct switch points, points on the boundary that bound a segment not touching the boundary.

This result is certainly false that for a $ {C^\infty }$ boundary. Indeed, even in $ {E^2}$, where our result is obvious for analytic boundaries, we can construct a $ {C^\infty }$ boundary so that the closure of the set of switch points is of positive measure.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1077783-8
PII: S 0002-9939(1991)1077783-8
Article copyright: © Copyright 1991 American Mathematical Society