201 research outputs found
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)
Combinatorial Depth Measures for Hyperplane Arrangements
Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion
that measures how good of a regression hyperplane a given query hyperplane is
with respect to a set of data points. Under projective duality, this can be
interpreted as a depth measure for query points with respect to an arrangement
of data hyperplanes. The study of depth measures for query points with respect
to a set of data points has a long history, and many such depth measures have
natural counterparts in the setting of hyperplane arrangements. For example,
regression depth is the counterpart of Tukey depth. Motivated by this, we study
general families of depth measures for hyperplane arrangements and show that
all of them must have a deep point. Along the way we prove a Tverberg-type
theorem for hyperplane arrangements, giving a positive answer to a conjecture
by Rousseeuw and Hubert from 1999. We also get three new proofs of the
centerpoint theorem for regression depth, all of which are either stronger or
more general than the original proof by Amenta, Bern, Eppstein, and Teng.
Finally, we prove a version of the center transversal theorem for regression
depth.Comment: To be presented at the 39th International Symposium on Computational
Geometry (SoCG 2023
Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidisc
We characterize the Carleson measures for the Dirichlet space on the bidisc,
hence also its multiplier space. Following Maz'ya and Stegenga, the
characterization is given in terms of a capacitary condition. We develop the
foundations of a bi-parameter potential theory on the bidisc and prove a Strong
Capacitary Inequality. In order to do so, we have to overcome the obstacle that
the Maximum Principle fails in the bi-parameter theory.Comment: 44 pages, 5 figures, title changed, minor editin
Discrete Geometry (hybrid meeting)
A number of important recent developments in various branches of
discrete geometry were presented at the workshop, which took place in
hybrid format due to a pandemic situation. The presentations
illustrated both the diversity of the area and its strong connections
to other fields of mathematics such as topology, combinatorics,
algebraic geometry or functional analysis. The open questions abound
and many of the results presented were obtained by young researchers,
confirming the great vitality of discrete geometry
Robust Optimal Power Distribution for Hyperthermia Cancer Treatment
We consider an optimization problem for spatial power distribution generated by an array of transmitting elements. Using ultrasound hyperthermia cancer treatment as a motivating example, the signal design problem consists of optimizing the power distribution across the tumor and healthy tissue regions, respectively. The models used in the optimization problem are, however, invariably subject to errors. To combat such unknown model errors, we formulate a robust signal design framework that can take the uncertainty into account using a worst-case approach. This leads to a semi-infinite programming (SIP) robust design problem, which we reformulate as a tractable convex problem that potentially has a wider range of applications
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