Rocky Mountain Journal of Mathematics
Boundary value problems
We give a complete and self-contained proof of a folklore theorem which says that in an Alexandrov space the distance between a point γ(t) on a geodesic γ and a compact set K is a right-differentiable function of t. Moreover, the value of this right-derivative is given by the negative cosine of the minimal angle between the geodesic and any shortest path to the compact set (Theorem 4.3). Our treatment serves as a general introduction to metric geometry and relies only on the basic elements, such as comparison triangles and upper angles.
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Fox, L. S., Oberly, P., & Veerman, J. J. P. (2021). One-sided derivative of distance to a compact set. Rocky Mountain Journal of Mathematics, 51(2). https://doi.org/10.1216/rmj.2021.51.491