61 research outputs found
Cutting the same fraction of several measures
We study some measure partition problems: Cut the same positive fraction of
measures in with a hyperplane or find a convex subset of
on which given measures have the same prescribed value. For
both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure
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
Convex Equipartitions via Equivariant Obstruction Theory
We describe a regular cell complex model for the configuration space
F(\R^d,n). Based on this, we use Equivariant Obstruction Theory to prove the
prime power case of the conjecture by Nandakumar and Ramana Rao that every
polygon can be partitioned into n convex parts of equal area and perimeter.Comment: Revised and improved version with extra explanations, 20 pages, 7
figures, to appear in Israel J. Mat
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