Name
Abstract | ||
|---|---|---|
Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well known that for any convex polygon there exist at least two empty and at least two full circles, i.e. at least four extremal circles. In 1990 Schatteman considered a generalization of this theorem for convex polytopes in d-dimensional Euclidean space. Namely, he claimed that there exist at least 2d extremal neighboring spheres for generic polytopes. His proof is based on the Bruggesser-Mani shelling method. In this paper, we show that there are certain gaps in Schatteman's proof. We also show that using the Bruggesser-Mani-Schatteman method it is possible to prove that there are at least d+1 extremal neighboring spheres. However, the existence problem of 2d extremal neighboring spheres is still open. |
Year | DOI | Venue |
|---|---|---|
2011 | 10.1016/j.disc.2010.10.012 | Discrete Mathematics |
Keywords | Field | DocType |
delaunay triangulation,shellability of complexes,four-vertex theorem,euclidean space,four vertex theorem,convex polytope | Discrete mathematics,Combinatorics,Polygon,Four-vertex theorem,Vertex (geometry),Convex polygon,Regular polygon,Euclidean space,Polytope,Convex polytope,Mathematics | Journal |
Volume | Issue | ISSN |
311 | 2-3 | Discrete Mathematics |
Citations | PageRank | References |
0 | 0.34 | 0 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Arseniy Akopyan | 1 | 3 | 2.81 |
| Alexey Glazyrin | 2 | 1 | 1.05 |
| Oleg R. Musin | 3 | 51 | 11.51 |
| Alexey Tarasov | 4 | 28 | 3.54 |