Geodesics in Euclidean space with analytic obstacle

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.