1,756 research outputs found
Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics
Using the geodesic distance on the -dimensional sphere, we study the
expected radius function of the Delaunay mosaic of a random set of points.
Specifically, we consider the partition of the mosaic into intervals of the
radius function and determine the expected number of intervals whose radii are
less than or equal to a given threshold. Assuming the points are not contained
in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of
the convex hull in , so we also get the expected number of
faces of a random inscribed polytope. We find that the expectations are
essentially the same as for the Poisson-Delaunay mosaic in -dimensional
Euclidean space. As proved by Antonelli and collaborators, an orthant section
of the -sphere is isometric to the standard -simplex equipped with the
Fisher information metric. It follows that the latter space has similar
stochastic properties as the -dimensional Euclidean space. Our results are
therefore relevant in information geometry and in population genetics
Weighted Poisson-Delaunay Mosaics
Slicing a Voronoi tessellation in with a -plane gives a
-dimensional weighted Voronoi tessellation, also known as power diagram or
Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay
mosaic to the radius of the smallest empty circumscribed sphere whose center
lies in the -plane gives a generalized discrete Morse function. Assuming the
Voronoi tessellation is generated by a Poisson point process in ,
we study the expected number of simplices in the -dimensional weighted
Delaunay mosaic as well as the expected number of intervals of the Morse
function, both as functions of a radius threshold. As a byproduct, we obtain a
new proof for the expected number of connected components (clumps) in a line
section of a circular Boolean model in $\mathbb{R}^n
IST Austria Thesis
The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the ÄŚech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's
Poisson–Delaunay Mosaics of Order k
The order-k Voronoi tessellation of a locally finite set ⊆ℝ decomposes ℝ into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold
The Beauty of Random Polytopes Inscribed in the 2-sphere
Consider a random set of points on the unit sphere in , which
can be either uniformly sampled or a Poisson point process. Its convex hull is
a random inscribed polytope, whose boundary approximates the sphere. We focus
on the case , for which there are elementary proofs and fascinating
formulas for metric properties. In particular, we study the fraction of acute
facets, the expected intrinsic volumes, the total edge length, and the distance
to a fixed point. Finally we generalize the results to the ellipsoid with
homeoid density.Comment: 18 pages, 4 figure
A step in the Delaunay mosaic of order k
Given a locally finite set ⊆ℝ and an integer ≥0, we consider the function :Del()→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space
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