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We study the existence of fair distributions when we have more guests than pieces to allocate, focusing on envy-free distributions among those who receive a piece. The conditions on the demand from the guests can be weakened from those of classic cake-cutting and rent-splitting results of Stromquist, Woodall, and Su. We extend existing variations of the cake-cutting problem with secretive guests and those that resist the removal of any sufficiently small set of guests. 

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