# Long geodesics on convex surfaces

## TL;DR

This paper explores the intrinsic geometry of convex surfaces, proving that only isosceles tetrahedra can contain arbitrarily long simple closed geodesics, thus linking geometric properties to specific polyhedral shapes.

## Contribution

It establishes a unique characterization of isosceles tetrahedra based on the existence of arbitrarily long simple closed geodesics on convex surfaces.

## Key findings

- Only isosceles tetrahedra can have arbitrarily long closed simple geodesics.
- Provides a geometric criterion linking geodesic length to polyhedral shape.
- Enhances understanding of intrinsic geometry of convex bodies.

## Abstract

We review the theory of intrinsic geometry of convex surfaces in the Euclidean space and prove the following theorem: if the surface of a convex body K contains arbitrary long closed simple geodesics, then K is an isosceles tetrahedron.