GEODESICS IN EUCLIDEAN-SPACE WITH ANALYTIC OBSTACLE

In this note we are concerned with the behavior of geodesics 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(x1, ...,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 infinity boundary. Indeed, even in E2 where our result is obvious for analytic boundaries, we can construct a C infinity boundary so that the closure of the set of switch points is of positive measure.