49,140 research outputs found
Piecewise-Linear Farthest-Site Voronoi Diagrams
Voronoi diagrams induced by distance functions whose unit balls are convex polyhedra are piecewise-linear structures. Nevertheless, analyzing their combinatorial and algorithmic properties in dimensions three and higher is an intriguing problem. The situation turns easier when the farthest-site variants of such Voronoi diagrams are considered, where each site gets assigned the region of all points in space farthest from (rather than closest to) it.
We give asymptotically tight upper and lower worst-case bounds on the combinatorial size of farthest-site Voronoi diagrams for convex polyhedral distance functions in general dimensions, and propose an optimal construction algorithm. Our approach is uniform in the sense that (1) it can be extended from point sites to sites that are convex polyhedra, (2) it covers the case where the distance function is additively and/or multiplicatively weighted, and (3) it allows an anisotropic scenario where each site gets allotted its particular convex distance polytope
Polyhedral Predictive Regions For Power System Applications
Despite substantial improvement in the development of forecasting approaches,
conditional and dynamic uncertainty estimates ought to be accommodated in
decision-making in power system operation and market, in order to yield either
cost-optimal decisions in expectation, or decision with probabilistic
guarantees. The representation of uncertainty serves as an interface between
forecasting and decision-making problems, with different approaches handling
various objects and their parameterization as input. Following substantial
developments based on scenario-based stochastic methods, robust and
chance-constrained optimization approaches have gained increasing attention.
These often rely on polyhedra as a representation of the convex envelope of
uncertainty. In the work, we aim to bridge the gap between the probabilistic
forecasting literature and such optimization approaches by generating forecasts
in the form of polyhedra with probabilistic guarantees. For that, we see
polyhedra as parameterized objects under alternative definitions (under
and norms), the parameters of which may be modelled and predicted.
We additionally discuss assessing the predictive skill of such multivariate
probabilistic forecasts. An application and related empirical investigation
results allow us to verify probabilistic calibration and predictive skills of
our polyhedra.Comment: 8 page
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
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