International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 63, Pages 3995-4006

Orthant spanning simplexes with minimal volume

Michele Elia

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.


A geometry problem is to find an (n1)-dimensional simplex in n 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 (2n1)-degree polynomial pn(t). The roots of pn(t) cannot be expressed using radicals when the coordinates of A are transcendental over , for 3n15, 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.