Orthant spanning simplexes with minimal volume
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...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203210401 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |