2023/12/16 by Vladimir Yu. Protasov, Protasov, Vladimir Yu.
Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Mathematics and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2312.10554
openalex publication_date 2023/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well-known that every isosceles tetrahedron (disphenoid) admits infinitely many simple closed geodesics on its surface. They can be naturally enumerated by pairs of co-prime integers n > m > 1 with two additional cases (1,0) and (1,1). The (n,m)-geodesic is a broken line with 4(n+m) vertices, its length tends to infinity as m→ ∞. Are there other polyhedra possessing this property? The answer depends on convexity. We give an elementary proof that among convex polyhedra only disphenoids admit arbitrarily long closed simple geodesics. For non-convex polyhedra, this is not true. We present a counterexample with the corresponding polyhedron being a union of seven equal cubes. Several open problems are formulated