A Volume Formual for Medial Sections of Simplices

Let Sd be a d-dimensional simplex in Rd , and let H be an affine hyperplane of Rd . We say that H is a medial hyperplane of Sd if the distance between H and any vertex of Sd is the same constant. The intersection of Sd and a medial hyperplane is called a medial section of Sd . In this paper we give a simple formula for the (d − 1)-volume of any medial section of Sd in terms of the lengths of the edges of Sd . This extends the result of Yetter [5] 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.