Lower bounds for the number of small convex k-holes
Let S be a set of n points in the plane in general position, that is, no three points of S are on a line. We consider an Erdos-type question on the least number h(k)(n) of convex k-holes in S, and give improved lower bounds on h(k)(n), for 3 <= k <= 5. Specifically, we show that h(3)(n) >=...
| Autores: | , , , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2014 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/26660 |
| Acceso en línea: | https://hdl.handle.net/2117/26660 https://dx.doi.org/10.1016/j.comgeo.2013.12.002 |
| Access Level: | acceso abierto |
| Palabra clave: | Discrete geometry Combinatorial geometry Empty convex polygon Erdos-type problem Counting PLANAR POINT SETS EMPTY POLYGONS THEOREM Geometria combinatòria Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria convexa i discreta |
| Sumario: | Let S be a set of n points in the plane in general position, that is, no three points of S are on a line. We consider an Erdos-type question on the least number h(k)(n) of convex k-holes in S, and give improved lower bounds on h(k)(n), for 3 <= k <= 5. Specifically, we show that h(3)(n) >= n(2) - 32n/7 + 22/7, h(4)(n) >= n(2)/2 - 9n/4 - o(n), and h(5)(n) >= 3n/4 - o(n). We further settle several questions on sets of 12 points posed by Dehnhardt in 1987. (C) 2013 Elsevier B.V. All rights reserved. |
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