2,133 research outputs found
Polyhedral Gauss-Bonnet theorems and valuations
The Gauss-Bonnet theorem for a polyhedron (a union of finitely many compact
convex polytopes) in -dimensional Euclidean space expresses the Euler
characteristic of the polyhedron as a sum of certain curvatures, which are
different from zero only at the vertices of the polyhedron. This note suggests
a generalization of these polyhedral vertex curvatures, based on valuations,
and thus obtains a variety of polyhedral Gauss-Bonnet theorems
Intersection probabilities and kinematic formulas for polyhedral cones
For polyhedral convex cones in , we give a proof for the conic
kinematic formula for conic curvature measures, which avoids the use of
characterization theorems. For the random cones defined as typical cones of an
isotropic random central hyperplane arrangement, we find probabilities for
non-trivial intersection, either with a fixed cone, or for two independent
random cones of this type
A Brunn-Minkowski theory for coconvex sets of finite volume
Let be a closed convex cone in , pointed and with interior
points. We consider sets of the form , where
is a closed convex set. If has finite volume (Lebesgue
measure), then is called a -coconvex set. The family of -coconvex
sets is closed under the addition defined by . We develop first steps of a
Brunn--Minkowski theory for -coconvex sets, which relates this addition to
the notion of volume. In particular, we establish the equality conditions for a
Brunn--Minkowski type inequality (with reversed inequality sign), introduce
mixed volumes and their integral representations, and prove a Minkowski-type
uniqueness theorem for -coconvex sets with equal surface area measures.Comment: The paper has been expanded by adding Minkowski type existence
theorems for surface area measures and cone-volume measure
Second moments related to Poisson hyperplane tessellations
We consider the typical cell of a stationary Poisson hyperplane tessellation
in d-dimensional Euclidean space. It is well known that the expected vertex
number of the typical cell is independent of the directional distribution of
the hyperplane process. We give sharp bounds for the variance of this vertex
number, showing, in particular, that the maximum of the variance is attained if
and only if the distribution of the process is rotation invariant with respect
to a suitable scalar product.
The employed representation of the second moment of the vertex number is a
special case of formulas providing the covariance matrix for the random vector
whose components are the total k-face contents of the typical cell. In the
isotropic case, such formulas were first obtained by R.E. Miles. We give a more
elementary proof and extend the formulas to general directional distributions.Comment: 11 page
Reflections of planar convex bodies
It is proved that every convex body in the plane has a point such that the
union of the body and its image under reflection in the point is convex. If the
body is not centrally symmetric, then it has, in fact, three affinely
independent points with this property.Comment: 7 page
Conic support measures
The conic support measures localize the conic intrinsic volumes of closed
convex cones in the same way as the support measures of convex bodies localize
the intrinsic volumes of convex bodies. In this note, we extend the `Master
Steiner formula' of McCoy and Tropp, which involves conic intrinsic volumes, to
conic support measures. Then we prove H\"{o}lder continuity of the conic
support measures with respect to the angular Hausdorff metric on convex cones
and a metric on conic support measures which metrizes the weak convergence
The polytopes in a Poisson hyperplane tessellation
For a stationary Poisson hyperplane tessellation in ,
whose directional distribution satisfies some mild conditions (which hold in
the isotropic case, for example), it was recently shown that with probability
one every combinatorial type of a simple -polytope is realized infinitely
often by the polytopes of . This result is strengthened here: with
probability one, every such combinatorial type appears among the polytopes of
not only infinitely often, but with positive density.Comment: 7 page
Rotation covariant local tensor valuations on convex bodies
For valuations on convex bodies in Euclidean spaces, there is by now a long
series of characterization and classification theorems. The classical template
is Hadwiger's theorem, saying that every rigid motion invariant, continuous,
real-valued valuation on convex bodies in is a linear
combination of the intrinsic volumes. For tensor-valued valuations, under the
assumptions of isometry covariance and continuity, there is a similar
classification theorem, due to Alesker. Also for the local extensions of the
intrinsic volumes, the support, curvature and area measures, there are
analogous characterization results, with continuity replaced by weak
continuity, and involving an additional assumption of local determination. The
present authors have recently obtained a corresponding characterization result
for local tensor valuations, or tensor-valued support measures (generalized
curvature measures), of convex bodies in . The covariance assumed
there was with respect to the group of orthogonal transformations.
This was suggested by Alesker's observation, according to which in dimensions
, the weaker assumption of covariance does not yield more
tensor valuations. However, for tensor-valued support measures, the distinction
between proper and improper rotations does make a difference. The present paper
considers, therefore, the local tensor valuations sharing the previously
assumed properties, but with covariance replaced by
covariance, and provides a complete classification. New tensor valued support
measures appear only in dimensions two and three
Local tensor valuations
The local Minkowski tensors are valuations on the space of convex bodies in
Euclidean space with values in a space of tensor measures. They generalize at
the same time the intrinsic volumes, the curvature measures and the isometry
covariant Minkowski tensors that were introduced by McMullen and characterized
by Alesker. In analogy to the characterization theorems of Hadwiger and
Alesker, we give here a complete classification of all locally defined tensor
measures on convex bodies that share with the local Minkowski tensors the basic
geometric properties of isometry covariance and weak continuity
covariant local tensor valuations on polytopes
The Minkowski tensors are valuations on the space of convex bodies in
with values in a space of symmetric tensors, having additional
covariance and continuity properties. They are extensions of the intrinsic
volumes, and as these, they are the subject of classification theorems, and
they admit localizations in the form of measure-valued valuations. For these
local tensor valuations, restricted to convex polytopes, a classification
theorem has been proved recently, under the assumption of isometry covariance,
but without any continuity assumption. This characterization result is extended
here, replacing the covariance under orthogonal transformations by invariance
under proper rotations only. This yields additional local tensor valuations on
polytopes in dimensions two and three, but not in higher dimensions. They are
completely classified in this paper
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