A Note on Lower Bounds for Colourful Simplicial Depth
The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construc...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2013-01-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | http://www.mdpi.com/2073-8994/5/1/47 |
| _version_ | 1811186136261853184 |
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| author | Antoine Deza Tamon Stephen Feng Xie |
| author_facet | Antoine Deza Tamon Stephen Feng Xie |
| author_sort | Antoine Deza |
| collection | DOAJ |
| description | The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is ⌈(d+1)2 /2 ⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14. |
| first_indexed | 2024-04-11T13:41:50Z |
| format | Article |
| id | doaj.art-3d47f16a1dc841acb829df847b5cb7fa |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T13:41:50Z |
| publishDate | 2013-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-3d47f16a1dc841acb829df847b5cb7fa2022-12-22T04:21:14ZengMDPI AGSymmetry2073-89942013-01-0151475310.3390/sym5010047A Note on Lower Bounds for Colourful Simplicial DepthAntoine DezaTamon StephenFeng XieThe colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is ⌈(d+1)2 /2 ⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14.http://www.mdpi.com/2073-8994/5/1/47colourful simplicial depthColourful Carath&#233odory Theoremdiscrete geometrypolyhedracombinatorial symmetry |
| spellingShingle | Antoine Deza Tamon Stephen Feng Xie A Note on Lower Bounds for Colourful Simplicial Depth Symmetry colourful simplicial depth Colourful Carath&#233 odory Theorem discrete geometry polyhedra combinatorial symmetry |
| title | A Note on Lower Bounds for Colourful Simplicial Depth |
| title_full | A Note on Lower Bounds for Colourful Simplicial Depth |
| title_fullStr | A Note on Lower Bounds for Colourful Simplicial Depth |
| title_full_unstemmed | A Note on Lower Bounds for Colourful Simplicial Depth |
| title_short | A Note on Lower Bounds for Colourful Simplicial Depth |
| title_sort | note on lower bounds for colourful simplicial depth |
| topic | colourful simplicial depth Colourful Carath&#233 odory Theorem discrete geometry polyhedra combinatorial symmetry |
| url | http://www.mdpi.com/2073-8994/5/1/47 |
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