485 research outputs found
Small Strong Epsilon Nets
Let P be a set of n points in . A point x is said to be a
centerpoint of P if x is contained in every convex object that contains more
than points of P. We call a point x a strong centerpoint for a
family of objects if is contained in every object that contains more than a constant fraction of points of P. A
strong centerpoint does not exist even for halfspaces in . We
prove that a strong centerpoint exists for axis-parallel boxes in
and give exact bounds. We then extend this to small strong
-nets in the plane and prove upper and lower bounds for
where is the family of axis-parallel
rectangles, halfspaces and disks. Here represents the
smallest real number in such that there exists an
-net of size i with respect to .Comment: 19 pages, 12 figure
Polychromatic Coloring for Half-Planes
We prove that for every integer , every finite set of points in the plane
can be -colored so that every half-plane that contains at least
points, also contains at least one point from every color class. We also show
that the bound is best possible. This improves the best previously known
lower and upper bounds of and respectively. We also show
that every finite set of half-planes can be colored so that if a point
belongs to a subset of at least of the half-planes then
contains a half-plane from every color class. This improves the best previously
known upper bound of . Another corollary of our first result is a new
proof of the existence of small size \eps-nets for points in the plane with
respect to half-planes.Comment: 11 pages, 5 figure
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