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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| Format: | Article |
| Language: | English |
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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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| _version_ | 1832562704642998272 |
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| author | Michele Elia |
| author_facet | Michele Elia |
| author_sort | Michele Elia |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-da213a2a2f1241c39df97bcfe9744915 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2003-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-da213a2a2f1241c39df97bcfe97449152025-02-03T01:22:05ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252003-01-012003633995400610.1155/S0161171203210401Orthant spanning simplexes with minimal volumeMichele Elia0Dipartimento di Elettronica, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, ItalyA 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.http://dx.doi.org/10.1155/S0161171203210401 |
| spellingShingle | Michele Elia Orthant spanning simplexes with minimal volume International Journal of Mathematics and Mathematical Sciences |
| title | Orthant spanning simplexes with minimal volume |
| title_full | Orthant spanning simplexes with minimal volume |
| title_fullStr | Orthant spanning simplexes with minimal volume |
| title_full_unstemmed | Orthant spanning simplexes with minimal volume |
| title_short | Orthant spanning simplexes with minimal volume |
| title_sort | orthant spanning simplexes with minimal volume |
| url | http://dx.doi.org/10.1155/S0161171203210401 |
| work_keys_str_mv | AT micheleelia orthantspanningsimplexeswithminimalvolume |