Change of polytope volumes under M\"{o}bius transformations and the circumcenter of mass

TL;DR

This paper explores how the volumes of polytopes change under Möbius transformations and introduces a triangulation-independent definition of the circumcenter of mass for simplicial polytopes.

Contribution

It provides a new definition of the circumcenter of mass that does not depend on triangulation and analyzes volume changes under Möbius transformations.

Findings

Volume of polytopes varies predictably under Möbius transformations.

A triangulation-independent definition of the circumcenter of mass is established.

Insights into geometric invariants under Möbius transformations.

Abstract

The circumcenter of mass of a simplicial polytope $P$ is defined as follows: triangulate $P$, assign to each simplex its circumcenter taken with weight equal to the volume of the simplex, and then find the center of mass of the resulting system of point masses. The so obtained point is independent of the triangulation. The aim of the present note is to give a definition of the circumcenter of mass that does not rely on a triangulation. To do so we investigate how volumes of polytopes change under M\"{o}bius transformations.