On non-onesided M-complete vector systems
The notion of //-convexity is a generalized convexity notion with many metrical and combinatorial applications (e.g., in distance geometry, combinatorial geometry, Minkowski geometry, and abstract convexity), and Hconvex sets are simply defined with the help of a finite or infinite system H of unit...
Elmentve itt :
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| Dokumentumtípus: | Cikk |
| Megjelent: |
Bolyai Institute, University of Szeged
Szeged
2008
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| Sorozat: | Acta scientiarum mathematicarum
74 No. 1-2 |
| Kulcsszavak: | Matematika |
| Tárgyszavak: | |
| Online Access: | http://acta.bibl.u-szeged.hu/16242 |
| Tartalmi kivonat: | The notion of //-convexity is a generalized convexity notion with many metrical and combinatorial applications (e.g., in distance geometry, combinatorial geometry, Minkowski geometry, and abstract convexity), and Hconvex sets are simply defined with the help of a finite or infinite system H of unit vectors in Euclidean n-space. In [7], [8], and [9] we investigated non-onesided, so-called M-complete systems of unit vectors and some of their applications in combinatorial geometry. In particular, we established a condition under which the vector (or Minkowski) sum of any two H-convex sets is again H-convex, and conditions for //-separability of H-convex sets. In both cases the notion of M-completeness, defined for the vector systems H , plays the key role. Here we study properties of maximal non-onesided, M - complete vector systems H and H in the unit sphere S n_1 , which means that any non-onesided, M-complete vector system containing them coincides with n_1 . On the other hand, we prove for closed systems, which are symmetric with respect to the origin, that the systems H and H are also universal, i.e., under some natural condition every non-onesided, M-complete vector system distinct from S n _ 1 is contained in H or in H. Some examples illustrate the obtained results. |
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| Terjedelem/Fizikai jellemzők: | 297-313 |
| ISSN: | 0001-6969 |