Viviani Polytopes and Fermat Points

Abstract: Given a set of oriented hyperplanes $\mathcal{P}={p_1, ..., p_k}$ in $\mathbb{R}<sup>n$, define $v(P)$ for any point $P\in\mathbb{R}<sup>n$ 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 $\mathbb{R}<sup>n$. Finally, we discuss historically the full content of Viviani's theorem.