International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 63, Pages 3995-4006
doi:10.1155/S0161171203210401

Orthant spanning simplexes with minimal volume

Abstract

A geometry problem is to find an (n−1)-dimensional simplex in ℝn of minimal volume with vertices on the positive coordinate axes, and constrained to pass through a given point A in the first orthant. In this paper, it is shown that the optimal simplex is identified by the only positive root of a (2n−1)-degree polynomial pn(t). The roots of pn(t) cannot be expressed using radicals when the coordinates of A are transcendental over ℚ, for 3≤n≤15, and supposedly for every n. Furthermore, limited to dimension 3, parametric representations are given to points A to which correspond triangles of minimal area with integer vertex coordinates and area.