A Volume Formual for Medial Sections of Simplices

Abstract

Let Sd be a d-dimensional simplex in Rd, and let H be an affine hyperplane of Rd. We say that H is a hyperplane of Sd if the distance between H and any vertex of Sd is the same constant. The intersection of Sd and a hyperplane is called a section of Sd. In this paper we give a simple formula for the (d-1)-volume of any section of Sd in terms of the lengths of the edges of Sd. This extends the result of Yetter from the three-dimensional case to arbitrary dimension. We also show that a generalization of the obtained formula measures the volume of the intersection of some analogously chosen medial affine subspace of Rd and the simplex.