arXiv subjects

Jan Rataj

Publications and source records attributed to Jan Rataj.

At least 19 recordsLinked to original sources

arXiv2025

A characterization of the local structure of two-dimensional sets with positive reach

The main result of the article is a complete characterization of the local structure of two-dimensional sets with positive reach in $R^d$. We also present a more elementary proof of a recent result of A. Lytchak which describes for $k\leq d$ the local structure of $k$-dimensional sets with positive reach $A$ in $R^d$ at points where the tangent cone of $A$ is $k$-dimensional. As an easy corollary of our and Lytchak's results we obtain a characterization of compact two-dimensional sets with positive reach in $R^d$. Our method also shows that, for any set $A\subset R^d$ with positive reach, the set of points at which the tangent cone of $A$ is $k$-dimensional is locally contained in a $k$-dimensional $C^{1,1}$ surface. As a consequence we obtain that if $1\leq k<d$, and $A$ is $k$-dimensional, it can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces.

math.MG↗

arXiv2025

Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals

We consider a class of random self-similar fractals based on code trees which includes random recursive, homogeneous and V-variable fractals and many more. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to 0. Moreover, integral representations are derived for these limits which recover and extend those known in the deterministic case and certain random cases. Results on the mean Minkowski content are included as a special case and shown to hold under weaker geometric assumptions.

math.PR↗

arXiv2025

On the (outer) Minkowski content with lower-dimensional structuring element

Given a convex body $Q$ (structuring element) and a set $A$ in a Euclidean space, we consider the $Q$-Minkowski content of $A$. It is defined as the usual isotropic Minkowski content of $A$, but where the Euclidean ball is replaced by $Q$. When $Q$ is full-dimensional, the existence of the $Q$-Minkowski content can be assured by a sufficient condition which was stated by Ambrosio, Fusco and Pallara in the isotropic case. If $Q$ is not full-dimensional, we show that a weaker condition is sufficient for this purpose. We also consider the outer $Q$-Minkowski content of $A$ yielding the anisotropic perimeter of $A$ and we find a sufficient condition for its existence. Finally, we present an example of a set in three-dimensional Euclidean space, which does not admit the isotropic outer Minkwski content, but it admits the outer $Q$-Minkowski content for all two-dimensional disks $Q$.

math.MG↗

arXiv2023

On volume and surface area of parallel sets. II. Surface measures and (non-)differentiability of the volume

We prove that at differentiability points $r_0>0$ of the volume function of a compact set $A\subset {\mathbb R}^d$ (associating to $r$ the volume of the $r$-parallel set of $A$), the surface area measures of $r$-parallel sets of $A$ converge weakly to the surface area measure of the $r_0$-parallel set as $r\to r_0$. We further study the question which sets of parallel radii can occur as sets of non-differentiability points of the volume function of some compact set. We provide a full characterization for dimensions $d=1$ and $2$.

math.MG↗

arXiv2022

Mean Lipschitz-Killing curvatures for homogeneous random fractals

Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the Uniform Strong Open Set Condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to zero. Moreover, integral representations are derived for these limits which extend those known in the deterministic case.

math.PR↗

arXiv2021

Mean Euler characteristic of stationary random closed sets

The translative intersection formula of integral geometry yields an expression for the mean Euler characteristic of a stationary random closed set intersected with a fixed observation window. We formulate this result in the setting of sets with positive reach and using flag measures which yield curvature measures as marginals. As an application, we consider excursion sets of stationary random fields with $C^{1,1}$ realizations, in particular, stationary Gaussian fields, and obtain results which extend those known from the literature.

math.PR↗

arXiv2019

Smallness of the set of critical values of distance functions in two-dimensional Euclidean and Riemannian spaces

We study how small is the set of critical values of the distance function from a compact (resp. closed) set in the plane or in a connected complete two-dimensional Riemannian manifold. We show that for a compact set, the set of critical values is compact and Lebesgue null (which is a known result) and that it has "locally" (away from 0) bounded sum of square roots of lengths of gaps (components of the complement). In the planar case, these conditions of local smallness are shown to be optimal. These results improve and generalize those of Fu (1985) and of our earlier paper from 2012. We also find an optimal condition for the smallness of the whole set of critical values of a planar compact set.

math.MG↗

arXiv2017

Flag representations of mixed volumes and mixed functionals of convex bodies

Mixed volumes $V(K_1,\dots, K_d)$ of convex bodies $K_1,\dots ,K_d$ in Euclidean space $\mathbb{R}^d$ are of central importance in the Brunn-Minkowski theory. Representations for mixed volumes are available in special cases, for example as integrals over the unit sphere with respect to mixed area measures. More generally, in Hug-Rataj-Weil (2013) a formula for $V(K [n], M[d-n])$, $n\in \{1,\dots ,d-1\}$, as a double integral over flag manifolds was established which involved certain flag measures of the convex bodies $K$ and $M$ (and required a general position of the bodies). In the following, we discuss the general case $V(K_1[n_1],\dots , K_k[n_k])$, $n_1+\cdots +n_k=d$, and show a corresponding result involving the flag measures $Ω_{n_1}(K_1;\cdot),\dots, Ω_{n_k}(K_k;\cdot)$. For this purpose, we first establish a curvature representation of mixed volumes over the normal bundles of the bodies involved. We also obtain a corresponding flag representation for the mixed functionals from translative integral geometry and a local version, for mixed (translative) curvature measures.

math.MG↗

arXiv2017

Dilation volumes of sets of finite perimeter

This paper analyzes the first order behavior (that is, the right sided derivative) of the volume of the dilation $A\oplus tQ$ as $t$ converges to zero. Here $A$ and $Q$ are subsets of $n$-dimensional Euclidean space, $A$ has finite perimeter and $Q$ is finite. If $Q$ consists of two points only, $x$ and $x+u$, say, this derivative coincides up to sign with the directional derivative of the covariogram of $A$ in direction $u$. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of $A$. We extend this result to finite sets $Q$ and use it to determine the derivative of the contact distribution function with finite structuring element of a stationary random set at zero. The proofs are based on approximation of the characteristic function of $A$ by smooth functions of bounded variation and showing corresponding formulas for them.

math.PR↗

arXiv2017

On the structure of WDC sets

WDC sets in ${\mathbb R}^d$ were recently defined as sublevel sets of DC functions (differences of convex functions) at weakly regular values. They form a natural and substantial generalization of sets with positive reach and still admit the definition of curvature measures. Using results on singularities of convex functions, we obtain regularity results on the boundaries of WDC sets. In particular, the boundary of a compact WDC set can be covered by finitely many DC surfaces. More generally, we prove that any compact WDC set $M$ of topological dimension $k\leq d$ can be decomposed into the union of two sets, one of them being a $k$-dimensional DC manifold open in $M$, and the other can be covered by finitely many DC surfaces of dimension $k-1$. We also characterize locally WDC sets among closed Lipschitz domains and among lower-dimensional Lipschitz manifolds. Finally, we find a full characterization of locally WDC sets in the plane.

math.MG↗

arXiv2016

Mixed curvature measures of translative integral geometry

The curvature measures of a set $X$ with singularities are measures concentrated on the normal bundle of $X$, which describe the local geometry of the set $X$. For given finitely many convex bodies or, more generally, sets with positive reach, the translative integral formula for curvature measures relates the integral mean of the curvature measures of the intersections of the given sets, one fixed and the others translated, to the mixed curvature measures of the given sets. In the case of two sets of positive reach, a representation of these mixed measures in terms of generalized curvatures, defined on the normal bundles of the sets, is known. For more than two sets, a description of mixed curvature measures in terms of rectifiable currents has been derived previously. Here we provide a representation of mixed curvatures measures of sets with positive reach based on generalized curvatures. The special case of convex polyhedra is treated in detail.

math.MG↗

arXiv2016

On the structure of sets with positive reach

We give a complete characterization of compact sets with positive reach (=proximally $C^1$ sets) in the plane and of one-dimensional sets with positive reach in ${\mathbb R}^d$. Further, we prove that if $\emptyset \neq A\subset{\mathbb R}^d$ is a set of positive reach of topological dimension $0< k \leq d$, then $A$ has its "$k$-dimensional regular part" $\emptyset \neq R \subset A$ which is a $k$-dimensional "uniform" $C^{1,1}$ manifold open in $A$ and $A\setminus R$ can be locally covered by finitely many $(k-1)$-dimensional DC surfaces. We also show that if $A \subset {\mathbb R}^d$ has positive reach, then $\partial A$ can be locally covered by finitely many semiconcave hypersurfaces.

math.MG↗

arXiv2015

Kinematic formulas for sets defined by differences of convex functions

Two of the authors have defined the class $ WDC(M)$ as the class of all subsets of a smooth manifold $M$ that may be expressed in local coordinates as certain sublevel sets of DC (differences of convex) functions. If $M$ is Riemanian and $G$ is a group of isometries acting transitively on the sphere bundle $SM$, we define the invariant curvature measures of compact \WDC~ subsets of $M$, and show that pairs of such subsets are subject to the array of kinematic formulas known to apply to smoother sets. Restricting to the case $(M, G) = (\mathbb R^d, \overline{SO(d)})$, this extends and subsumes Federer's theory of sets with positive reach in an essential way. The key technical point is equivalent to a sharpening of a classical theorem of Ewald, Larman, and Rogers characterizing the dimension of the set of directions of line segments lying in the boundary of a given convex body.

math.DG↗

arXiv2015

A flag representation of projection functions

The $k$th projection function $v_k(K,\cdot)$ of a convex body $K\subset {\mathbb R}^d, d\ge 3,$ is a function on the Grassmannian $G(d,k)$ which measures the $k$-dimensional volume of the projection of $K$ onto members of $G(d,k)$. For $k=1$ and $k=d-1$, simple formulas for the projection functions exist. In particular, $v_{d-1}(K,\cdot)$ can be written as a spherical integral with respect to the surface area measure of $K$. Here, we generalize this result and prove two integral representations for $v_k(K,\cdot), k=1,\dots,d-1$, over flag manifolds. Whereas the first representation generalizes a result of Ambartzumian (1987), but uses a flag measure which is not continuous in $K$, the second representation is related to a recent flag formula for mixed volumes by Hug, Rataj and Weil (2013) and depends continuously on $K$.

math.MG↗

arXiv2014

Legendrian cycles and curvatures

Properties of general Legendrian cycles $T$ acting in ${\mathbb R}^d\times S^{d-1}$ are studied. In particular, we give short proofs for certain uniqueness theorems with respect to the projections on the first and second component of such currents: In general, $T$ is determined by its restriction to the Gauss curvature form - this result goes back to J. Fu - and in the full-dimensional case also by the restriction to the surface area form. As a tool a version of the Constancy theorem for Lipschitz submanifolds is shown.

math.DG↗

arXiv2014

Random sets of finite perimeter

An approach to modelling random sets with locally finite perimeter as random elements in the corresponding subspace of $L^1$ functions is suggested. A Crofton formula for flat sections of the perimeter is shown. Finally, random processes of particles with finite perimeter are introduced and it is shown that their union sets are random sets with locally finite perimeter.

math.PR↗

arXiv2013

Normal cycles and curvature measures of sets with d.c. boundary

We show that for every compact domain in a Euclidean space with d.c. (delta-convex) boundary there exists a unique Legendrian cycle such that the associated curvature measures fulfil a local version of the Gauss-Bonnet formula. This was known in dimensions two and three and was open in higher dimensions. In fact, we show this property for a larger class of sets including also lower-dimensional sets. We also describe the local index function of the Legendrian cycles and we show that the associated curvature measures fulfill the Crofton formula.

math.DG↗

arXiv2013

A product integral representation of mixed volumes of two convex bodies

The Brunn-Minkowski theory relies heavily on the notion of mixed volumes. Despite its particular importance, even explicit representations for the mixed volumes of two convex bodies in Euclidean space are available only in special cases. Here we investigate a new integral representation of such mixed volumes, in terms of flag measures of the involved convex sets. A brief introduction to (extended) flag measures of convex bodies is also provided.

math.MG↗