A formula with volumes of five tetrahedra and discrete curvature

Abstract

Given five points in a three-dimensional euclidean space, one can consider five tetrahedra, using those points as vertices. We present a pentagon-like formula containing the product of three volumes of those tetrahedra in its l.h.s. and the product of the two remaining tetrahedron volumes in its r.h.s., as well as the derivative of the "discrete curvature" which arises when we slightly deform our euclidean space.