294 research outputs found
On Strong Centerpoints
Let be a set of points in and be a
family of geometric objects. We call a point a strong centerpoint of
w.r.t if is contained in all that
contains more than points from , where is a fixed constant. A
strong centerpoint does not exist even when is the family of
halfspaces in the plane. We prove the existence of strong centerpoints with
exact constants for convex polytopes defined by a fixed set of orientations. We
also prove the existence of strong centerpoints for abstract set systems with
bounded intersection
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
A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing
Motivated by an open problem from graph drawing, we study several
partitioning problems for line and hyperplane arrangements. We prove a
ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l
such that in both line sets, for both halfplanes delimited by l, there are
n^{1/2} lines which pairwise intersect in that halfplane, and this bound is
tight; a centerpoint theorem: for any set of n lines there is a point such that
for any halfplane containing that point there are (n/3)^{1/2} of the lines
which pairwise intersect in that halfplane. We generalize those results in
higher dimension and obtain a center transversal theorem, a same-type lemma,
and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This
is done by formulating a generalization of the center transversal theorem which
applies to set functions that are much more general than measures. Back to
Graph Drawing (and in the plane), we completely solve the open problem that
motivated our search: there is no set of n labelled lines that are universal
for all n-vertex labelled planar graphs. As a side note, we prove that every
set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar
graphs
An Optimal Generalization of the Colorful Carathéodory Theorem
International audienceThe Colorful Carathéodory theorem by Bárány (1982) states that given d + 1 sets of points in R d , the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d + 1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the Colorful Carathéodory theorem: given + 1 sets of points in R d and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a rainbow simplex intersecting C
Extending the Centerpoint Theorem to Multiple Points
The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set P of n points in R^d, there is a point c, not necessarily from P, such that each halfspace containing c contains at least n/(d+1) points of P. Such a point c is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set P. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median.
We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set Q of (few) points such that every halfspace that contains one point of Q contains a large fraction of the points of P and every halfspace that contains more of Q contains an even larger fraction of P. This setting is comparable to the well-studied concepts of weak epsilon-nets and weak epsilon-approximations, where it is stronger than the former but weaker than the latter. We show that for any point set of size n in R^d and for any positive alpha_1,...,alpha_k where alpha_1 <= alpha_2 <= ... <= alpha_k and for every i,j with i+j <= k+1 we have that (d-1)alpha_k+alpha_i+alpha_j <= 1, we can find Q of size k such that each halfspace containing j points of Q contains least alpha_j n points of P. For two-dimensional point sets we further show that for every alpha and beta with alpha <= beta and alpha+beta <= 2/3 we can find Q with |Q|=3 such that each halfplane containing one point of Q contains at least alpha n of the points of P and each halfplane containing all of Q contains at least beta n points of P. All these results generalize to the setting where P is any mass distribution. For the case where P is a point set in R^2 and |Q|=2, we provide algorithms to find such points in time O(n log^3 n)
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