A Volume Formual for Medial Sections of Simplices

Let S _d be a d-dimensional simplex in R ^d , and let H be an affine hyperplane of R ^d . We say that H is a medial hyperplane of S _d if the distance between H and any vertex of S _d is the same constant. The intersection of S _d and a medial hyperplane is called a medial section of S _d . In this paper we give a simple formula for the (d-1)-volume of any medial section of S _d in terms of the lengths of the edges of S _d . 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 R ^d and the simplex.