2,450 research outputs found
Mass Partitions via Equivariant Sections of Stiefel Bundles
We consider a geometric combinatorial problem naturally associated to the
geometric topology of certain spherical space forms. Given a collection of
mass distributions on , the existence of affinely independent
regular -fans, each of which equipartitions each of the measures, can in
many cases be deduced from the existence of a -equivariant
section of the Stiefel bundle over , where
is the Stiefel manifold of all orthonormal -frames in
or , and
is the corresponding unit sphere. For example, the
parallelizability of when , or implies that any
two masses on can be simultaneously bisected by each of
pairwise-orthogonal hyperplanes, while when or 4, the triviality of the
circle bundle over the standard Lens Spaces
yields that for any mass on , there exist a pair of
complex orthogonal regular -fans, each of which equipartitions the mass.Comment: 11 pages, final versio
Knaster's problem for -symmetric subsets of the sphere
We prove a Knaster-type result for orbits of the group in
, calculating the Euler class obstruction. Among the consequences
are: a result about inscribing skew crosspolytopes in hypersurfaces in , and a result about equipartition of a measures in
by -symmetric convex fans
Balanced Islands in Two Colored Point Sets in the Plane
Let be a set of points in general position in the plane, of which
are red and of which are blue. In this paper we prove that there exist: for
every , a convex set containing
exactly red points and exactly
blue points of ; a convex set containing exactly red points and exactly blue points of . Furthermore, we present
polynomial time algorithms to find these convex sets. In the first case we
provide an time algorithm and an time algorithm in the
second case. Finally, if is
small, that is, not much larger than , we improve the running
time to
- …