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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41353/1/454_2003_Article_15.pd