Geodesics in Euclidean Space with Analytic Obstacle

Felix Albrecht, I. D. Berg
1991 Proceedings of the American Mathematical Society  
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 xn = f(xx, ... , xn_x) 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 C°° boundary.
more » ... C°° boundary. Indeed, even in E , where our result is obvious for analytic boundaries, we can construct a C°°b oundary so that the closure of the set of switch points is of positive measure.
doi:10.2307/2048459 fatcat:oipxptvf4ngy7l55jbuf4q5ssq