International Journal of Mathematics and Mathematical Sciences (Jan 2003)

Orthant spanning simplexes with minimal volume

  • Michele Elia

DOI
https://doi.org/10.1155/S0161171203210401
Journal volume & issue
Vol. 2003, no. 63
pp. 3995 – 4006

Abstract

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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.