# Spherical centroid bodies

## TL;DR

This paper introduces the concept of spherical centroid bodies for centrally-symmetric convex bodies on the sphere, establishing their properties and proving a spherical version of the centroid inequality.

## Contribution

It defines spherical centroid bodies using geometric and probabilistic methods and proves a new spherical centroid inequality.

## Key findings

- Defined spherical centroid bodies via two approaches
- Proved fundamental properties of these bodies
- Established a spherical analogue of the centroid inequality

## Abstract

The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions - one geometric, the other probabilistic in nature - are given and shown to lead to the same objects. The geometric approach is then used to establish a number of basic properties of spherical centroid bodies, while the probabilistic approach inspires the proof of a spherical analogue of the classical polar Busemann-Petty centroid inequality.