Viviani Polytopes and Fermat Points

Given a set of oriented hyperplanes $\mathcal{P}=\{p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_k\}$ 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.