1,603 research outputs found
Envy-free cake division without assuming the players prefer nonempty pieces
Consider players having preferences over the connected pieces of a cake,
identified with the interval . A classical theorem, found independently
by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it
is possible to divide the cake into connected pieces and assign these
pieces to the players in an envy-free manner, i.e, such that no player strictly
prefers a piece that has not been assigned to her. One of these conditions,
considered as crucial, is that no player is happy with an empty piece. We prove
that, even if this condition is not satisfied, it is still possible to get such
a division when is a prime number or is equal to . When is at most
, this has been previously proved by Erel Segal-Halevi, who conjectured that
the result holds for any . The main step in our proof is a new combinatorial
lemma in topology, close to a conjecture by Segal-Halevi and which is
reminiscent of the celebrated Sperner lemma: instead of restricting the labels
that can appear on each face of the simplex, the lemma considers labelings that
enjoy a certain symmetry on the boundary
Piercing axis-parallel boxes
Let \F be a finite family of axis-parallel boxes in such that \F
contains no pairwise disjoint boxes. We prove that if \F contains a
subfamily \M of pairwise disjoint boxes with the property that for every
F\in \F and M\in \M with , either contains a
corner of or contains corners of , then \F can be
pierced by points. One consequence of this result is that if and
the ratio between any of the side lengths of any box is bounded by a constant,
then \F can be pierced by points. We further show that if for each two
intersecting boxes in \F a corner of one is contained in the other, then \F
can be pierced by at most points, and in the special case
where \F contains only cubes this bound improves to
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