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CUTTING A PART FROM MANY MEASURES
Published online by Cambridge University Press: 17 October 2019
Abstract
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
$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$.
- Type
- Research Article
- Information
- Creative Commons
- 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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