Viviani Polytopes and Fermat Points

TL;DR

This paper characterizes hyperplane configurations in Euclidean space where the sum of signed distances from any point is constant, linking this to Fermat points and revisiting Viviani's theorem historically.

Contribution

It provides a geometric characterization of hyperplane arrangements with constant signed distance sums and connects this to Fermat points in Euclidean space.

Findings

Characterization of hyperplanes with constant signed distance sums

Connection established between these hyperplanes and Fermat points

Historical discussion of Viviani's theorem

Abstract

Given a set of oriented hyperplanes $\mathcal{P}=\{p_1, ..., p_k\}$ in $\mathbb{R}^n$, define $v(P)$ for any point $P\in\mathbb{R}^n$ as the sum of the signed distances from $P$ to $p_1$,..., $p_k$. We give a simple geometric characterization of $\mathcal{P}$ so that $v$ is a constant. The characterization leads to a connection with the Fermat point of $k$ points in $\mathbb{R}^n$. Finally, we discuss historically the full content of Viviani's theorem.