The min-max width of spheres associated to the distance function
Differential Geometry
2025-03-18 v1
Abstract
What one obtains when the min-max methods for the distance function are applied on the space of pairs of points of a Riemannian two-sphere? This question is studied in details in the present article. We show that the associated min-max width do not always coincide with half of the length of a simple closed geodesic which is the union of two minimizing geodesics with the same endpoints. Therefore, it is a new geometric invariant. We study the structure of the set of minimizing geodesics joining a pair of points realizing the width, and relationships between this invariant and the diameter. The extrinsic case of an embedded Riemannian sphere is also considered.
Keywords
Cite
@article{arxiv.2503.12477,
title = {The min-max width of spheres associated to the distance function},
author = {Rafael Montezuma and Idalina Ribeiro},
journal= {arXiv preprint arXiv:2503.12477},
year = {2025}
}