Orthant spanning simplexes with minimal volume

A geometry problem is to find an ( n − 1 ) -dimensional simplex in ℝ 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 ( 2 n − 1 ) -degree polynomial p n ( t ) . The roots of p n ( 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.