Jan Rataj

Normal cycles of Lipschitz manifolds by approximation with parallel sets

Abstract It is shown that sufficiently close inner parallel sets and closures of the complements of outer parallel sets to a d-dimensional Lipschitz manifold in R d with boundary have locally positive reach and the normal cycle of the Lipschitz manifold can be defined as limit of normal cycles of the parallel sets in the flat seminorms for currents, provided that the normal cycles of the parallel set have locally bounded mass. The Gauss–Bonnet formula and principal kinematic formula are proved for these normal cycles. It is shown that locally finite unions of non-osculating sets with positive reach of full dimension, as well as the closures of their complements, admit such a definition of normal cycle.

Mixed curvature measures for sets of positive reach and a translative integral formula

Mixed curvature measures for sets of positive reach are introduced and a translative version of the principal kinematic formula from integral geometry is proved. This is an extension of a known result from convex geometry. Integral representations of the mixed curvature measures in various particular cases of dimension two and three are derived.

On Infinitesimal Increase of Volumes of Morphological Transforms

Let B (“black”) and W (“white”) be disjoint compact test sets in ℝ d , and consider the volume of all its simultaneous shifts keeping B inside and W outside a compact set A ⊂ ℝ d . If the union B ∪ W is rescaled by a factor tending to zero, then the rescaled volume converges to a value determined by the surface area measure of A and the support functions of B and W , provided that A is regular enough ( e.g. , polyconvex). An analogous formula is obtained for the case when the conditions B ⊂ A and W ⊂ A C are replaced by prescribed threshold volumes of B in A and W in A C . Applications in stochastic geometry are discussed. First, the hit distribution function of a random set with an arbitrary compact structuring element B is considered. Its derivative at 0 is expressed in terms of the rose of directions and B . An analogous result holds for the hit-or-miss function. Second, in a design based setting, different random digitizations of a deterministic set A are treated. It is shown how the number of configurations in such a digitization is related to the surface area measure of A as the lattice distance converges to zero.

Non-homogeneous distribution of internal stresses and cracks in poled PZT ceramics

Abstract The spatial distribution of intergranular cracks in poled coarse-grained PZT ceramics is found to be non-homogeneous with a higher density of cracks in the central region of the specimen and a lower density near the electrodes. As the cracks are formed by internal stresses (due to motion of 90° walls) it is also concluded that the distribution of internal stresses (even in fine-grained transducers without any cracks) is non-homogeneous on a macroscopic scale with the highest peak stress values in the central region.

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

Abstract 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 fulfill 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.

Random sets of finite perimeter

An approach to modelling random sets with locally finite perimeter as random elements in the corresponding subspace of L1 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.

Absolute curvature measures for unions of sets with positive reach

Absolute curvature measures for locally finite unions of sets with positive reach are introduced, extending the definition of Zahle [13] by taking into account the absolute value of the index function. It is shown that this definition differs from that of Matheron [5] and Schneider [12]. An intersection formula of Crofton type for absolute curvature measures is proved. The role of absolute curvature measures in geometric statistics is illustrated by an example.

A Translative Integral Formula for Absolute Curvature Measures

Absolute curvature measures are certain positive variants of curvature measures which are defined by means of the kinematic measure of flats of a given dimension touching the examined body. A kinematic flat section formula for absolute curvature measures was proved by Baddeley and Rother and Zähle. We present here a translative version of this formula.

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.

On estimation of the Euler number by projections of thin slabs

A stereological formula for the Euler number involving projections of the set in thin parallel slabs is considered. Sufficient conditions for the validity of this formula are derived.