arXiv subjects

Jan Rataj

Publications and source records attributed to Jan Rataj.

24 records · Page 2Linked to original sources

arXiv2011

Characterization of Minkowski measurability in terms of surface area

The $r$-parallel set to a set $A$ in Euclidean space consists of all points with distance at most $r$ from $A$. Recently, the asymptotic behaviour of volume and the surface area of parallel sets as $r$ tends to 0 has been studied and some general results regarding their relations have been established. In this paper we complete this picture. In particular, we show that a set is Minkowski measurable if and only if it is S-measurable, i.e. if its S-content is positive and finite, and that positivity and finiteness of lower and upper Minkowski content implies the same for the S-contents and vice versa. The results are formulated in the more general setting of Kneser functions. Furthermore, the relations between Minkowski and S-contents are studied for more general gauge functions. The results are also applied to simplify the proof of the Modified Weyl-Berry conjecture in dimension one.

math.CA↗

arXiv2010

On Volume and Surface Area of Parallel Sets

The r-parallel set to a set A in a Euclidean space consists of all points with distance at most r from A. We clarify the relation between the volume and the surface area of parallel sets and study the asymptotic behaviour of both quantities as r tends to 0. We show, for instance, that in general, the existence of a (suitably rescaled) limit of the surface area implies the existence of the corresponding limit for the volume, known as the Minkowski content. A full characterisation is obtained for the case of self-similar fractal sets. Applications to stationary random sets are discussed as well, in particular, to the trajectory of the Brownian motion.

math.CA↗

arXiv2010

Curvature densities of self-similar sets

For a large class of self-similar sets F in R^d analogues of the higher order mean curvatures of differentiable submanifolds are introduced, in particular, the fractal Gauss-type curvature. They are shown to be the densities of associated fractal curvature measures, which are all multiples of the corresponding Hausdorff measures on F, due to its self-similarity. This local approach based on ergodic theory for an associated dynamical system enables us to extend former global curvature results.

math.MG↗

arXiv2009

Properties of distance functions on convex surfaces and applications

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 $X\times X$ in the only natural extrinsic sense. An analogous result holds for the squared distance function $\dist^2(x,F)$ from a closed set $F \subset X$. Applications concerning $r$-boundaries (distance spheres) and the ambiguous locus (exoskeleton) of a closed subset of a convex surface are given.

math.MG↗

arXiv2009

Critical values and level sets of distance functions in Riemannian, Alexandrov and Minkowski spaces

Let $F \subset \R^n$ be a closed set and $n=2$ or $n=3$. S. Ferry (1975) proved that then, for almost all $r>0$, the level set (distance sphere, $r$-boundary) $S_r(F):= \{x \in \R^n: \dist(x,F) = r\}$ is a topological $(n-1)$-dimensional manifold. This result was improved by J.H.G. Fu (1985). We show that Ferry's result is an easy consequence of the only fact that the distance function $d(x)= \dist(x,F)$ is locally DC and has no stationary point in $\R^n\setminus F$. Using this observation, we show that Ferry's (and even Fu's) result extends to sufficiently smooth normed linear spaces $X$ with $\dim X \in \{2,3\}$ (e.g., to $\ell^p_n, n=2,3, p\geq 2$), which improves and generalizes a result of R. Gariepy and W.D. Pepe (1972). By the same method we also generalize Fu's result to Riemannian manifolds and improve a result of K. Shiohama and M. Tanaka (1996) on distance spheres in Alexandrov spaces.

math.MG↗

arXiv2009

Approximations of the Wiener sausage and its curvature measures

A parallel neighborhood of a path of a Brownian motion is sometimes called the Wiener sausage. We consider almost sure approximations of this random set by a sequence of random polyconvex sets and show that the convergence of the corresponding mean curvature measures holds under certain conditions in two and three dimensions. Based on these convergence results, the mean curvature measures of the Wiener sausage are calculated numerically by Monte Carlo simulations in two dimensions. The corresponding approximation formulae are given.

math.PR↗