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Full-Text Articles in Physical Sciences and Mathematics
One-Sided Derivative Of Distance To A Compact Set, Logan S. Fox, Peter Oberly, J. J. P. Veerman
One-Sided Derivative Of Distance To A Compact Set, Logan S. Fox, Peter Oberly, J. J. P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
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.