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On set covariance and three-point test sets

Jan Rataj — 2004

Czechoslovak Mathematical Journal

The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of d is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives of the characteristic...

Almost sure asymptotic behaviour of the r -neighbourhood surface area of Brownian paths

Ondřej HonzlJan Rataj — 2012

Czechoslovak Mathematical Journal

We show that whenever the q -dimensional Minkowski content of a subset A d exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in d , d 3 .

Properties of distance functions on convex surfaces and applications

Jan RatajLuděk Zajíček — 2011

Czechoslovak Mathematical Journal

If X is a convex surface in a Euclidean space, then the squared intrinsic distance function dist 2 ( x , y ) is DC (d.c., delta-convex) on X × X in the only natural extrinsic sense. An analogous result holds for the squared distance function dist 2 ( x , F ) from a closed set F X . Applications concerning r -boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface are given.

On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane

Tomáš MrkvičkaJan Rataj — 2009

Kybernetika

A method of estimation of intrinsic volume densities for stationary random closed sets in d based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the...

The iterated version of a translative integral formula for sets of positive reach

Rataj, Jan — 1997

Proceedings of the 16th Winter School "Geometry and Physics"

By taking into account the work of and [Geom. Dedicata 57, 259-283 (1995; Zbl 0844.53050)], and [Math. Nachr. 129, 67-80 (1986; Zbl 0602.52003)], [Math. Z. 205, 531-549 (1990; Zbl 0705.52006)], an integral formula is obtained here by using the technique of rectifiable currents.This is an iterated version of the principal kinematic formula for q sets of positive reach and generalized curvature measures.

Convergence of randomly oscillating point patterns to the Poisson point process

Jan RatajIvan SaxlKarol Pelikán — 1993

Applications of Mathematics

Oscillating point patterns are point processes derived from a locally finite set in a finite dimensional space by i.i.d. random oscillation of individual points. An upper and lower bound for the variation distance of the oscillating point pattern from the limit stationary Poisson process is established. As a consequence, the true order of the convergence rate in variation norm for the special case of isotropic Gaussian oscillations applied to the regular cubic net is found. To illustrate these theoretical...

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