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- J Rataj
- 2007

A class of subsets of R d which can be represented as locally nite unions of sets with positive reach is considered. It plays a role in PDE's on manifolds with singularities. For such a set, the unit normal cycle (determining the d ?1 curvature measures) is introduced as a (d ?1)-current supported by the unit normal bundle and its properties are… (More)

- Tomás Mrkvicka, Jan Rataj
- Kybernetika
- 2009

It is shown that sufficiently close outer and inner 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… (More)

- THIELE CENTRE, Markus Kiderlen, Jan Rataj, MARKUS KIDERLEN
- 2005

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… (More)

Closed Legendrian (d − 1)-dimensional locally rectifiable currents on the sphere bundle in R d are considered and the associated index functions are studied. A topological condition assuring the validity of a local version of the Gauss-Bonnet formula is established. The case of lower-dimensional Lipschitz submanifolds in R d and their associated normal… (More)

For parallel neighborhoods of the paths of the d–dimensional Brownian motion, so–called Wiener sausages, formulae for the expected surface area are given for any dimension d ≥ 2. It is shown by means of geometric arguments that the expected surface area is equal to the first derivative of the mean volume of the Wiener sausage with respect to its radius.

- J. Rataj
- 2005

The existence of mixed curvature measures of two sets in R d with positive reach introduced in 4] is discussed. An example shows that the non-osculating condition from 4] does not ensure the locally bounded variation of the mixed curvature measures. It is shown that if both sets have locally bounded tangential projections then their mixed curvature measure… (More)

- JAN RATAJ
- 2009

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 the ambiguous locus… (More)

- JAN RATAJ
- 2009

If X is a convex surface in a Euclidean space, then the squared (intrinsic) distance function dist 2 (x, y) is d.c. (DC, delta-convex) on X × X in the only natural extrinsic sense. For the proof we use semiconcavity (in an intrinsic sense) of dist 2 (x, y) on X × X if X is an Alexandrov space with nonnegative curvature. Applications concerning r-boundaries… (More)