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
Dipartimento di Elettronica, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy
Received 23 October 2002
Copyright © 2003 Michele Elia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.