The classical Crofton formula explains how intrinsic volumes of a convex body K in n-dimensional Euclidean space can be obtained from integrating a measurement function at sections of K with ...invariantly moved affine flats. Motivated by stereological applications, we present variants of Crofton's formula, where the flats are constrained to contain a fixed linear subspace L0, but are otherwise invariantly rotated. This main result generalizes a known rotational Crofton formula, which only covers the case dimL0=0. The proof combines a suitable Blaschke–Petkantschin formula with the classical Crofton formula. We also argue that our main result is best possible, in the sense that one cannot estimate intrinsic volumes of a set, based on lower-dimensional sections, other than those given by our result. Finally, we provide a proof for a well-established variant: an integral relation for vertical sections. Our formula is stated for intrinsic volumes of a given set, complementing the classical approach for Hausdorff measures.
Let
Ω
⊂
R
2
and let
L
⊂
Ω
be a one-dimensional set with finite length
L
=
|
L
|
. We are interested in minimizers of an energy functional that measures the size of a set projected onto itself in all ...directions: we are thus asking for sets that see themselves as little as possible (suitably interpreted). Obvious minimizers of the functional are subsets of a straight line but this is only possible for
L
≤
diam
(
Ω
)
. The problem has an equivalent formulation: the expected number of intersections between a random line and
L
depends only on the length of
L
(Crofton’s formula). We are interested in sets
L
that minimize the variance of the expected number of intersections. We solve the problem for convex
Ω
and slightly less than half of all values of
L
: there, a minimizing set is the union of copies of the boundary and a line segment.
We explore the notion of Santaló point for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the norm is C1, and in the case where unit ball and convex body ...coincide, we prove existence and uniqueness. When the normed space has a smooth positively curved unit ball, we exhibit a dual Santaló point expressed as an average of centroids of projections of the dual body.
Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz–Killing curvature integrals of a submanifold with corners, to be computed by integrating ...the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.
A lower bound on opaque sets Kawamura, Akitoshi; Moriyama, Sonoko; Otachi, Yota ...
Computational geometry : theory and applications,
July 2019, 2019-07-00, Volume:
80
Journal Article
Peer reviewed
Open access
It is proved that the total length of any set of countably many rectifiable curves whose union meets all straight lines that intersect the unit square U is at least 2.00002. This is the first ...improvement on the lower bound of 2 known since 1964. A similar bound is proved for all convex sets U other than a triangle.
Poisson processes in the space of
(
d
-
1
)
-dimensional totally geodesic subspaces (hyperplanes) in a
d
-dimensional hyperbolic space of constant curvature
-
1
are studied. The
k
-dimensional ...Hausdorff measure of their
k
-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the
k
-dimensional Hausdorff measure of the
k
-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all
d
≥
2
, it is shown that in case (ii) the central limit theorem holds for
d
∈
{
2
,
3
}
and fails if
d
≥
4
and
k
=
d
-
1
or if
d
≥
7
and for general
k
. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.
Methods to estimate surface areas of geometric objects in
3D are well known. A number of these methods are of Monte Carlo type,
and some are based on the Cauchy–Crofton formula from integral ...geometry.
Employing this formula requires the generation of sets of random lines that
are uniformly distributed in 3D. One model to generate sets of random lines
that are uniformly distributed in 3D is called the tangent model (see
). In this paper, we present an extension of this model to
higher dimensions, and we examine its performance by estimating hypersurface
areas of
-ellipsoids. Then we apply this method to estimate surface areas
of hypersurfaces defined by Fermat-type varieties of even degree.
Crofton’s formula of integral geometry evaluates the total motion invariant measure of the set of
k
-dimensional planes having nonempty intersection with a given convex body. This note deals with ...motion invariant measures on sets of pairs of hyperplanes or lines meeting a convex body. Particularly simple results are obtained if, and only if, the given body is of constant width in the first case, and of constant brightness in the second case.
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