On the representation of finite convex geometries with convex sets
Very recently Richter and Rogers proved that any convex geometry can be represented by a family of convex polygons in the plane. We shall generalize their construction and obtain a wide variety of convex shapes for representing convex geometries. We present an Erdős-Szekeres type obstruction, which...
Elmentve itt :
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| Dokumentumtípus: | Cikk |
| Megjelent: |
Bolyai Institute, University of Szeged
Szeged
2017
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| Sorozat: | Acta scientiarum mathematicarum
83 No. 1-2 |
| Kulcsszavak: | Matematika, Konvex halmazok geometriája |
| Tárgyszavak: | |
| doi: | 10.14232/actasm-017-502-z |
| Online Access: | http://acta.bibl.u-szeged.hu/48932 |
| Tartalmi kivonat: | Very recently Richter and Rogers proved that any convex geometry can be represented by a family of convex polygons in the plane. We shall generalize their construction and obtain a wide variety of convex shapes for representing convex geometries. We present an Erdős-Szekeres type obstruction, which answers a question of Czédli negatively, that is general convex geometries cannot be represented with ellipses in the plane. Moreover, we shall prove that one cannot even bound the number of common supporting lines of the pairs of the representing convex sets. In higher dimensions we prove that all convex geometries can be represented with ellipsoids. |
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| Terjedelem/Fizikai jellemzők: | 301-312 |
| ISSN: | 0001 6969 |