The hamburger theorem

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F18%3A10366176" target="_blank" >RIV/00216208:11320/18:10366176 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1016/j.comgeo.2017.06.012" target="_blank" >http://dx.doi.org/10.1016/j.comgeo.2017.06.012</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.comgeo.2017.06.012" target="_blank" >10.1016/j.comgeo.2017.06.012</a>

Result language
angličtina
Original language name
The hamburger theorem
Original language description
We generalize the ham sandwich theorem to d+1 measures on R^d as follows. Let mu_1, mu_2, ..., mu_{d+1} be absolutely continuous finite Borel measures on R^d. Let omega_i=mu_i(R^d) for i in [d+1], omega=min{omega_i; i in [d+1]} and assume that sum_{j=1}^{d+1} omega_j=1. Assume that omega_i <= 1/d for every i in [d+1]. Then there exists a hyperplane h such that each open halfspace H defined by h satisfies mu_i(H) <= (sum_{j=1}^{d+1} mu_j(H))/d for every i in [d+1] and sum_{j=1}^{d+1} mu_j(H) >= min{1/2, 1 - d*omega} >= 1/(d+1). As a consequence we obtain that every (d+1)-colored set of nd points in R^d such that no color is used for more than n points can be partitioned into n disjoint rainbow (d-1)-dimensional simplices.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA14-14179S" target="_blank" >GA14-14179S: Algorithmic, structural and complexity aspects of configurations in the plane</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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