{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:29:50Z","timestamp":1760243390530,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2013,1,7]],"date-time":"2013-01-07T00:00:00Z","timestamp":1357516800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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 \u2265 4 is \u2308(d+1)2 \/2 \u2309. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14.<\/jats:p>","DOI":"10.3390\/sym5010047","type":"journal-article","created":{"date-parts":[[2013,1,7]],"date-time":"2013-01-07T11:07:53Z","timestamp":1357556873000},"page":"47-53","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Note on Lower Bounds for Colourful Simplicial Depth"],"prefix":"10.3390","volume":"5","author":[{"given":"Antoine","family":"Deza","sequence":"first","affiliation":[{"name":"Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4K1, Canada"}]},{"given":"Tamon","family":"Stephen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada"}]},{"given":"Feng","family":"Xie","sequence":"additional","affiliation":[{"name":"Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4K1, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2013,1,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/0012-365X(82)90115-7","article-title":"A generalization of Carath\u00e9odory\u2019s theorem","volume":"40","year":"1982","journal-title":"Discret. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1007\/s00454-006-1233-3","article-title":"Colourful simplicial depth","volume":"35","author":"Deza","year":"2006","journal-title":"Discret. Comput. Geom."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1137\/050643039","article-title":"Quadratically many colorful simplices","volume":"21","year":"2007","journal-title":"SIAM J. Discret. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1007\/s10878-008-9149-x","article-title":"A quadratic lower bound for colourful simplicial depth","volume":"16","author":"Stephen","year":"2008","journal-title":"J. Comb. Opt."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1007\/s00454-010-9291-y","article-title":"More colourful simplices","volume":"45","author":"Deza","year":"2011","journal-title":"Discret. Comput. Geom."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1214\/aos\/1176347507","article-title":"On a notion of data depth based on random simplices","volume":"18","author":"Liu","year":"1990","journal-title":"Ann. Statist."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1007\/s00039-010-0073-8","article-title":"Singularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry","volume":"20","author":"Gromov","year":"2010","journal-title":"Geom. Funct. Anal."},{"key":"ref_8","unstructured":"Matou\u0161ek, J., and Wagner, U. On Gromov\u2019s method of selecting heavily covered points. Available online: http:\/\/arxiv.org\/abs\/1102.3515."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1007\/s00454-011-9332-1","article-title":"A simpler proof of the Boros-F\u00fcredi-B\u00e1r\u00e1ny-Pach-Gromov theorem","volume":"47","author":"Karasev","year":"2012","journal-title":"Discret. Comput. Geom."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1007\/s00454-012-9419-3","article-title":"A new lower bound based on Gromov\u2019s method of selecting heavily covered points","volume":"48","author":"Mach","year":"2012","journal-title":"Discret. Comput. Geom."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"550","DOI":"10.1287\/moor.22.3.550","article-title":"Colourful linear programming and its relatives","volume":"22","author":"Onn","year":"1997","journal-title":"Math. Oper. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2166","DOI":"10.1016\/j.dam.2008.01.016","article-title":"The colourful feasibility problem","volume":"156","author":"Deza","year":"2008","journal-title":"Discret. Appl. Math."},{"key":"ref_13","unstructured":"Custard, G., Deza, A., Stephen, T., and Xie, F. (2011, January 10\u201312). Small octahedral systems. Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, Toronto, Ontario, Canada."},{"key":"ref_14","unstructured":"B\u00e1r\u00e1ny, I. Hungarian Academy of Sciences, Budapest, Hungary. Personal communication."},{"key":"ref_15","unstructured":"Deza, A., Stephen, T., and Xie, F. Computational lower bounds for colourful simplicial depth. Available online: http:\/\/arxiv.org\/abs\/1210.7621."},{"key":"ref_16","unstructured":"Xie, F. Python code for octrahedral system computation. Available online: http:\/\/optlab.mcmaster.ca\/om\/csd\/."},{"key":"ref_17","unstructured":"Deza, A., Meunier, F., and Sarrabezolles, P. A combinatorial approach to colourful simplicial depth. Available online: http:\/\/arxiv.org\/abs\/\/1212.4720."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Schulz, A., and T\u00f3th, C.D. (2013). The union of colorful simplices spanned by a colored point set. Comput. Geom., in press.","DOI":"10.1016\/j.comgeo.2012.01.006"},{"key":"ref_19","first-page":"147","article-title":"Geometric measures of data depth","volume":"Volume 72","author":"Aloupis","year":"2006","journal-title":"Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/5\/1\/47\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:44:07Z","timestamp":1760219047000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/5\/1\/47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1,7]]},"references-count":19,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2013,3]]}},"alternative-id":["sym5010047"],"URL":"https:\/\/doi.org\/10.3390\/sym5010047","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2013,1,7]]}}}