Geodesics in Euclidean space with analytic obstacle

Abstract

In this note we are concerned with the behavior of geodesies in Euclidean n 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 , … , x n − 1 ) {x_n} = f({x_1}, \ldots ,{x_{n - 1}}) for a real analytic function f 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 ∞ {C^\infty } boundary. Indeed, even in E 2 {E^2} , where our result is obvious for analytic boundaries, we can construct a C ∞ {C^\infty } boundary so that the closure of the set of switch points is of positive measure.