17 research outputs found
KKM type theorems with boundary conditions
We consider generalizations of Gale's colored KKM lemma and Shapley's KKMS
theorem. It is shown that spaces and covers can be much more general and the
boundary KKM rules can be substituted by more weaker boundary assumptions.Comment: 13 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1406.6672 by other author
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
Multilabeled versions of Sperner's and Fan's lemmas and applications
We propose a general technique related to the polytopal Sperner lemma for
proving old and new multilabeled versions of Sperner's lemma. A notable
application of this technique yields a cake-cutting theorem where the number of
players and the number of pieces can be independently chosen. We also prove
multilabeled versions of Fan's lemma, a combinatorial analogue of the
Borsuk-Ulam theorem, and exhibit applications to fair division and graph
coloring.Comment: 21 pages, 2 figure
Generalized Rental Harmony
Rental Harmony is the problem of assigning rooms in a rented house to tenants
with different preferences, and simultaneously splitting the rent among them,
such that no tenant envies the bundle (room+price) given to another tenant.
Different papers have studied this problem under two incompatible assumptions:
the miserly tenants assumption is that each tenant prefers a free room to a
non-free room; the quasilinear tenants assumption is that each tenant
attributes a monetary value to each room, and prefers a room of which the
difference between value and price is maximum. This note shows how to adapt the
main technique used for rental harmony with miserly tenants, using Sperner's
lemma, to a much more general class of preferences, that contains both miserly
and quasilinear tenants as special cases. This implies that some recent results
derived for miserly tenants apply to this more general preference class too.Comment: Generalized all results to "compensable tenants" - a class that
contains both miserly and quasilinear tenant
A Polytopal Generalization of Sperner\u27s Lemma
We prove the following conjecture of Atanassov (Studia Sci. Math. Hungar.32 (1996), 71–74). Let T be a triangulation of a d-dimensional polytope P with n vertices v1, v2,…,vn. Label the vertices of T by 1,2,…,n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if vj is on F. Then there are at least n−d full dimensional simplices of T, each labelled with d+1 different labels. We provide two proofs of this result: a non-constructive proof introducing the notion of a pebble set of a polytope, and a constructive proof using a path-following argument. Our non-constructive proof has interesting relations to minimal simplicial covers of convex polyhedra and their chamber complexes, as in Alekseyevskaya (Discrete Math.157 (1996), 15–37) and Billera et al. (J. Combin. Theory Ser. B57 (1993), 258–268)
Colorful Borsuk--Ulam theorems and applications
We prove a colorful generalization of the Borsuk--Ulam theorem and derive
colorful consequences from it, such as a colorful generalization of the ham
sandwich theorem. Even in the uncolored case this specializes to a
strengthening of the ham sandwich theorem, which given an additional condition,
contains a result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated
measures as a special case. We prove a colorful generalization of Fan's
antipodal sphere covering theorem, we derive a short proof of Gale's colorful
KKM theorem, and we prove a colorful generalization of Brouwer's fixed point
theorem. Our results also provide an alternative between Radon-type
intersection results and KKM-type covering results. Finally, we prove colorful
Borsuk--Ulam theorems for higher symmetry.Comment: 15 page