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CUTTING A PART FROM MANY MEASURES

Part of: Discrete geometry Homology and cohomology theories Fiber spaces and bundles Real and complex geometry

Published online by Cambridge University Press:  17 October 2019

PAVLE V. M. BLAGOJEVIĆ Open the ORCID record for PAVLE V. M. BLAGOJEVIĆ [Opens in a new window] ,
NEVENA PALIĆ ,
PABLO SOBERÓN Open the ORCID record for PABLO SOBERÓN [Opens in a new window]  and
GÜNTER M. ZIEGLER Open the ORCID record for GÜNTER M. ZIEGLER [Opens in a new window]
PAVLE V. M. BLAGOJEVIĆ
Affiliation:
Institut für Mathematik, FU Berlin, Arnimallee 2, 14195 Berlin, Germany; blagojevic@math.fu-berlin.de, nevenapalic@gmail.com, ziegler@math.fu-berlin.de Mathematical Institute SASA, Knez Mihailova 36, 11000 Beograd, Serbia
NEVENA PALIĆ
Affiliation:
Institut für Mathematik, FU Berlin, Arnimallee 2, 14195 Berlin, Germany; blagojevic@math.fu-berlin.de, nevenapalic@gmail.com, ziegler@math.fu-berlin.de
PABLO SOBERÓN
Affiliation:
Baruch College, City University of New York, One Bernard Baruch Way, New York, NY 10010, USA; pablo.soberon-bravo@baruch.cuny.edu
GÜNTER M. ZIEGLER
Affiliation:
Institut für Mathematik, FU Berlin, Arnimallee 2, 14195 Berlin, Germany; blagojevic@math.fu-berlin.de, nevenapalic@gmail.com, ziegler@math.fu-berlin.de

Abstract

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Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^{d}$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint.

We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ to be greater than $d$. Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geqslant 2$, $d\geqslant 2$, $c\geqslant d$ with $m\geqslant n(c-d)+d$ are integers, and $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^{d}$, we prove that there exists a partition of $\mathbb{R}^{d}$ into $n$ convex pieces which equiparts the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{d-1}$, and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$.

MSC classification

Primary: 52C35: Arrangements of points, flats, hyperplanes 51M20: Polyhedra and polytopes; regular figures, division of spaces
Secondary: 55R20: Spectral sequences and homology of fiber spaces 55N25: Homology with local coefficients, equivariant cohomology
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e37
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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