On a Formula Relating Volumes of Medial Sections of Simplices

摘要

Talata's medial sections of simplices can be classified into families according to how many vertices of the simplex lie on one side of the hyperplane determining the section. Defining a (k,l)-medial section to be one in which k vertices lie on one side and l=d+1-k on the other, we show that for any d-simplex the ratio of the sum of the squared volumes of the (k,l)-medial sections to the sum of the squared volumes of the (kappa,lambda)-medial sections is independent of the geometry of the simplex, depending only on d, k, and kappa. We give a formula for the ratio.