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The book describes how curvature measures can be introduced for
certain classes of sets with singularities in Euclidean spaces. Its
focus lies on sets with positive reach and some extensions, which
include the classical polyconvex sets and piecewise smooth
submanifolds as special cases. The measures under consideration
form a complete system of certain Euclidean invariants. Techniques
of geometric measure theory, in particular, rectifiable currents
are applied, and some important integral-geometric formulas are
derived. Moreover, an approach to curvatures for a class of
fractals is presented, which uses approximation by the rescaled
curvature measures of small neighborhoods. The book collects
results published during the last few decades in a nearly
comprehensive way.
Stochastic geometry, based on current developments in geometry,
probability and measure theory, makes possible modeling of two- and
three-dimensional random objects with interactions as they appear
in the microstructure of materials, biological tissues,
macroscopically in soil, geological sediments etc. In combination
with spatial statistics it is used for the solution of practical
problems such as the description of spatial arrangements and the
estimation of object characteristics. A related field is
stereology, which makes possible inference on the structures, based
on lower-dimensional observations. Unfolding problems for particle
systems and extremes of particle characteristics are studied. The
reader can learn about current developments in stochastic geometry
with mathematical rigor on one hand and find applications to real
microstructure analysis in natural and material sciences on the
other hand.
Stochastic geometry, based on current developments in geometry,
probability and measure theory, makes possible modeling of two- and
three-dimensional random objects with interactions as they appear
in the microstructure of materials, biological tissues,
macroscopically in soil, geological sediments etc. In combination
with spatial statistics it is used for the solution of practical
problems such as the description of spatial arrangements and the
estimation of object characteristics. A related field is
stereology, which makes possible inference on the structures, based
on lower-dimensional observations. Unfolding problems for particle
systems and extremes of particle characteristics are studied. The
reader can learn about current developments in stochastic geometry
with mathematical rigor on one hand and find applications to real
microstructure analysis in natural and material sciences on the
other hand.
The book describes how curvature measures can be introduced for
certain classes of sets with singularities in Euclidean spaces. Its
focus lies on sets with positive reach and some extensions, which
include the classical polyconvex sets and piecewise smooth
submanifolds as special cases. The measures under consideration
form a complete system of certain Euclidean invariants. Techniques
of geometric measure theory, in particular, rectifiable currents
are applied, and some important integral-geometric formulas are
derived. Moreover, an approach to curvatures for a class of
fractals is presented, which uses approximation by the rescaled
curvature measures of small neighborhoods. The book collects
results published during the last few decades in a nearly
comprehensive way.
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