  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.
