Relationship between Euclidean, Lobachevskian (hyperbolic), and billiard metrics and its application to a billiard problem in $R^d$

A relationship between the three metrics — Billiard, Euclidean, and Lobachevskian (Hyperbolic) — is established in the article. This relationship is applied to a billiard problem on generalized diagonals of a Euclidean multidimensional convex polyhedron.