A simple sum for simplices

TL;DR

The paper presents a vector identity for points in Euclidean space that reveals a parity-dependent sum of simplex volumes, showing a zero-sum property for odd and even dimensions.

Contribution

It introduces a new vector identity relating points in Euclidean space to the sum of simplex volumes, highlighting parity-based volume sum properties.

Findings

Sum of signed volumes is zero for odd n.

Alternating sum of volumes is zero for even n.

Provides a new geometric vector identity.

Abstract

We give a vector identity for $n+2$ points in $\mathbb R^n$. It follows as a corollary that when $n$ is odd the sum of the signed volumes of the $n$-simplices is zero, and when $n$ is even, the alternating sum of the signed volumes is zero.