# Integral of distance function on compact Riemannian manifolds

## TL;DR

This paper proves that on compact Riemannian manifolds with certain curvature conditions, the integral of the distance function has a universal lower bound related to the manifold's diameter, volume, and dimension.

## Contribution

It establishes a new lower bound for the integral of the distance function on compact Riemannian manifolds under curvature assumptions, linking geometric quantities.

## Key findings

- Lower bound depends on diameter, volume, and dimension
- Bound holds under specific curvature conditions
- Results applicable to a class of Riemannian manifolds

## Abstract

In this paper we show that, under some curvature assumptions the integral of distance function on a compact Riemannian manifold is bounded below by the product of diameter, volume and a constant only depending on the dimension.