Name
Abstract | ||
|---|---|---|
We generalize the ham sandwich theorem to d+1 measures on Rd as follows. Let μ1,μ2,…,μd+1 be absolutely continuous finite Borel measures on Rd. Let ωi=μi(Rd) for i∈[d+1], ω=min{ωi;i∈[d+1]} and assume that ∑j=1d+1ωj=1. Assume that ωi≤1/d for every i∈[d+1]. Then there exists a hyperplane h such that each open halfspace H defined by h satisfies μi(H)≤(∑j=1d+1μj(H))/d for every i∈[d+1] and ∑j=1d+1μj(H)≥min{1/2,1−dω}≥1/(d+1). As a consequence we obtain that every (d+1)-colored set of nd points in Rd such that no color is used for more than n points can be partitioned into n disjoint rainbow (d−1)-dimensional simplices. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1016/j.comgeo.2017.06.012 | Computational Geometry |
Keywords | Field | DocType |
Borsuk–Ulam theorem,Ham sandwich theorem,Hamburger theorem,Absolutely continuous Borel measure,Colored point set | Ham sandwich theorem,Discrete mathematics,Combinatorics,Disjoint sets,Absolute continuity,Hyperplane,Borsuk–Ulam theorem,Mathematics | Journal |
Volume | ISSN | Citations |
68 | 0925-7721 | 3 |
PageRank | References | Authors |
0.49 | 2 | 2 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Mikio Kano | 1 | 548 | 99.79 |
| Jan Kyncl | 2 | 6 | 1.24 |