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 -space with a smooth obstacle. Our principal result is that if the obstacle is locally analytic, that is, locally of the form for a real analytic function , 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 boundary. Indeed, even in , where our result is obvious for analytic boundaries, we can construct a boundary so that the closure of the set of switch points is of positive measure.

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DOI:
https://doi.org/10.1090/S0002-9939-1991-1077783-8

Article copyright:
© Copyright 1991
American Mathematical Society