Reach of orbits of the isotropy representations of Riemannian symmetric spaces

In this paper, we study the geometric properties of orbits of isotropy representations of Riemannian semisimple symmetric spaces. The main result of the paper is the construction of a combinatorial-geometric model for calculating the critical radii of orbits of isotropy representations. According to G. Federer, the critical radius of a subset of a Euclidean space (the English term reach) is the largest size of a tubular neighborhood of the subset in which each element has a unique nearest element in the specified subset. This metric invariant of subsets in various metric spaces has recently been actively studied in connection with applications in probability theory and topological data analysis.