Viviani Polytopes and Fermat Points

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“…Theorem 2: If a prism is created by the orthogonal translation in space of an equilateral polygon, equi-angled polygon or a 2n-gon with opposite sides parallel, then the sum of the distances from a point P to the faces of the prism is constant. But this theorem is a very special case of the much more general Theorem I in [2]:…”

“…On '3D Generalisations of Viviani's Theorem': Li Zhou writes: The article [1] did not refer to [2], which has generalised Viviani's theorem to any fmite dimension n. In particular, we read in [1]:…”

“…Then the sum of the signed distances from any point P to the hyperplanes is constant if, and only if, n, + ... + n, = O. A reference to [2] was also made at Wikipedia [3] in December, 2010. on noting that Li, (l) =~(3) and Li3 (-1) = -i~(3).…”