(1) Department of Mathematics, The University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, USA;(2) Present address: Department of Geometry, Eötvös University, P.O. Box 120, Budapest, Hungary
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 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 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.