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
$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$
.