# A diameter gap for quotients of the unit sphere

## TL;DR

This paper proves a universal lower bound or zero for the diameter of quotient spaces resulting from isometric group actions on high-dimensional unit spheres, revealing a fundamental geometric property.

## Contribution

It establishes a dimension-independent diameter gap for quotients of the unit sphere under isometric group actions, a novel geometric insight.

## Key findings

- Quotients have diameter zero or at least a positive constant
- The diameter gap is independent of the sphere's dimension
- Provides a universal property for group actions on spheres

## Abstract

We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.