New Crofton formulae derived from existing ones.
problem Generalizing Crofton formulae for products.
method Calculations in the ring of normal densities.
result Generalizations of Crofton formulae in terms of mixed Riemannian volume.
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48 results for “Crofton formula”
Developed new Crofton formulas for pseudo-Riemannian spaces.
problem Computing volumes and curvature integrals in pseudo-Riemannian space forms.
method Introduced Crofton formulas using distributions and Alesker's Radon transform.
result Explicit Crofton formulas for all isometry-invariant valuations on pseudo-Riemannian spaces.
Unified approach to Crofton and Hurwitz integral formulas for convex sets.
problem Integral formulas for visual angle of convex sets.
method Unified integral formula tool for visual angle functions.
result Unified upper and lower bounds for considered integrals.
Study -invariant valuations and derive Crofton formulas.
problem Classify and describe -invariant valuations.
method Explicit formulas for signatures, functorial properties, and Crofton distributions.
result Every -invariant valuation is given by an invariant Crofton formula.
The paper interprets and proves integral formulas for visual angles.
problem Integral formulas of visual angles.
method Integral Geometry approach to Crofton, Hurwitz, and Masotti formulas.
result New simpler proofs of integral formulas.
A solution of Hilberts fourth problem lead to integral equation of the type generalized cosine transform. The present paper considers the solution that integral equation by integral geometry methods and propose an inversion formula for reconstruction of Crofton measures from projective smooth Finsler metrics in R3.
We generalize the Fenchel theorem to strong spacelike (which means that the tangent vector and the curvature vector span a spacelike 2-plane at each point) closed curves with index 1 in the 3-dimensional Lorentz space, showing that the total curvatures must be less than or equal to . A similar generalization of the…
New valuations on contact manifolds generalize Euclidean integral geometry.
problem Understanding curvature in contact geometry.
method Reinterpreting valuations from Riemannian to contact manifolds.
result Contact manifolds have canonical families of generalized valuations.
Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.
problem Finding surface area of arbitrary submanifolds in R^n.
method Defining natural projected areas and volumes, deriving a recursive formula.
result Derived a new surface area formula that coincides with Crofton's and De Jong's formulas.
This note generalizes the visual angle to convex sets in 3D space.
problem Analyzing geometric properties of convex sets in 3D space.
method Generalizing the visual angle to convex sets in Euclidean space and expressing geometric quantities in terms of integrals of functions related to the solid angle.
result Invariant quantities of the original convex set can be expressed by integrals of functions related to the solid angle.
Existence of smooth valuations on subspaces is shown for certain conditions.
problem Existence of smooth valuations on subspaces with given restrictions.
method Analyzing compatibility and using recursive descriptions of the cosine transform.
result Compatibility is sufficient for extensibility in certain regimes.
New integral-geometric formulae derived from normal densities ring.
problem Smooth versions of BKK theorem.
method Algorithm based on ring of normal densities.
result Smooth versions of BKK theorem obtained.
Estimates average number of common zeros of Laplacian eigenfunctions on manifolds.
problem Estimating the average number of common zeros of Laplacian eigenfunctions on compact Riemannian manifolds.
method Application of Crofton's formula for the sphere.
result Proves an estimate for the average number of common zeros of eigenfunctions, showing it does not exceed a specific expression.
We prove new kinematic formulas for tensor valuations and simplify previously known Crofton formulas by using the recently developed algebraic theory of translation invariant valuations. The heart of the paper is the computation of the Alesker-Fourier transform on the large class of spherical valuations, which is achie…
New formulas for measuring geometric properties of definable sets.
problem Measuring geometric properties of definable sets in a non-standard setting.
method Proved two kinematic formulas integrating on .
result Generalized Cauchy-Crofton and infinitesimal linear kinematic formulas.
Length of parallel curvesmath.DG
We prove that the length difference between a closed periodic curve and its parallel curve at a sufficiently small distance is proportional to the rotation index. As an application, the rotation index of a curve could be estimated by means of Cauchy-Crofton formula.
We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives …
We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length distribution for an ideal polygon in the hyperbolic plane. The analogous results …
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 fulfil 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 sh…
Average zeros of Finsler functions equals mixed symplectic volume of ellipsoids.
problem Counting isolated common zeros of Finsler functions.
method Construction of ring of normal densities and Crofton formula generalization.
result Average number of zeros equals mixed symplectic volume of Finsler ellipsoids.
We construct a class of Finsler metrics in three-dimensional space such that all their geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This shows that if the Hausdorff measure is used as notion of volume on Finsler spaces, then totally geodesic submanifolds are not necessarily…
The paper proposes a new way to approximate Riemannian metrics using discrete wall systems.
problem Approximating Riemannian metrics and proving geometric conjectures.
method Discretization of metrics using walls and triangulations.
result The discrete filling area conjecture is equivalent to Gromov's original conjecture.
Develops calculus for random submanifolds using zonoids.
problem Calculating properties of random submanifolds defined by zero sets of vector fields.
method Defines zonoid sections and uses them to compute expected volumes and currents.
result Establishes new inequalities and formulas for random submanifolds.
We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is prove…
The article develops a mathematical theory for uniformly distributing points on surfaces.
problem Creating representative point clouds on surfaces and higher-dimensional manifolds.
method Using Cauchy Crofton formula and its generalizations from Integral Geometry.
result A rigorous mathematical theory for point clouds on various manifolds.
The paper explores connections between perimeter, area, and visual angle of convex sets.
problem Understanding geometric properties of convex sets through visual angle and related measurements.
method Establishing universal formulas and characterizing convex sets of constant width.
result Crofton's formula is the unique universal formula relating visual angle, length, and area.
In order to study large variations or fluctuations of finite or infinite sequences (time series), we bring to light an 1868 paper of Crofton and the (Cauchy-)Crofton theorem. After surveying occurrences of this result in the literature, we introduce the inconstancy of a sequence and we show why it seems more pertinent …
Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space. In particular, one can attempt this in R^n equipped with the metric derived from the p-norm. This has, in effect, been investigated intensively for 1<p<\infty, but not for p=1. We show that integral …
The paper evaluates integrals of planes and their relation to convex set angles.
problem Integrals of invariant measures of pairs of planes in .
method Expressing integrals in terms of visual angle functions of convex sets.
result Evaluation of a Crofton-type inequality deficit.
A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations.
The width of a curve in Euclidean space is the infimum of the distances between all pairs of parallel hyperplanes which bound , while its inradius is the supremum of the radii of all spheres which are contained in the convex hull of and are disjoint from . We use a mixture of topological and…
Cauchy's work on integral geometry, centers of curvature, and other applications of infinitesimalsmath.HO
Cauchy used infinitesimals in differential geometry and integral geometry.
problem Applying infinitesimals in differential and integral geometry.
method Using infinitesimals as numbers in differential and integral geometry.
result Valid application of infinitesimals in geometric probability, differential geometry, elasticity, and Dirac delta functions.
We define a new notion of total curvature, called net total curvature, for finite graphs embedded in Rn, and investigate its properties. Two guiding principles are given by Milnor's way of measuring the local crookedness of a Jordan curve via a Crofton-type formula, and by considering the double cover of a given graph …
We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algorithms in order to increase their convergence speed. We consider algorithms that sample from a probability density conditioned on a manifold…
The study improves bounds on space waists using advanced geometry techniques.
problem Understanding the waist of different geometric spaces.
method Advanced geometric techniques, including Borsuk-Ulam-Crofton method, Hausdorff measure.
result Established new waist bounds for various spaces.
We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related w…
Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
Derives an integral formula for G2-structures.
problem Calculating properties of G2-structures.
method Applies an integral formula for G-structures to G2.
result Derives an integral formula relating curvatures and quadratic invariants.
Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
Paper proves a fixed point formula and applies it to a new proof of Harish-Chandra's character formula.
problem Proving a fixed point formula for equivariant indices of elliptic differential operators.
method Fixed point formula for proper actions by connected semisimple Lie groups on manifolds.
result New proof of Harish-Chandra's character formula for discrete series representations.
Application of a unified Kenmotsu-type formula for surfaces in Euclidean or Lorentzian three-spacemath.DG
Unified formula for surfaces in Euclidean or Lorentzian 3-space.
problem Describe surfaces in Euclidean or Lorentzian 3-space.
method Unified Kenmotsu-type formula for surfaces in Euclidean or Lorentzian 3-space.
result Unified single equation for Kenmotsu-type formulas in Euclidean and Lorentzian 3-space.
Two tropical gluing formulas help calculate Gromov-Witten invariants.
problem Calculating Gromov-Witten invariants of symplectic manifolds.
method Tropical geometry applied to exploded manifolds.
result Generalizes existing formulas for Gromov-Witten invariants.
Paper derives trace formula for magnetic Laplacian at zero energy.
problem Trace formula for magnetic Laplacian at zero energy.
method Generalizes Gutzwiller trace formula, focuses on zero energy level.
result Derives trace formula at zero energy level.
The Gauss formula is extended to various Laplacians on submanifolds.
problem Deriving formulas for Laplacians on submanifolds.
method Extending the Gauss formula to different types of Laplacians.
result Formulas for various Laplacians on submanifolds.
Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin classmath.AT
The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…
Note on new cancellation formulas for manifolds.
problem Generalizing anomaly cancellation formulas to manifolds.
method Proving new (a, b) type cancellation formulas and using transgression.
result Obtained characteristic forms with modularity properties.
The study identifies types of manifolds using variational formulas and integral-differential formulas.
problem Identifying specific types of manifolds based on curvature properties.
method Established variational formulas for Ricci curvature bounds and used them to identify manifolds.
result Constant curvature, Einstein, and Ricci parallel manifolds identified with specific formulas.
The main result of the present paper is a coincidence formula for foliated manifolds. To prove this we establish Kuenneth formula, Poincare duality and intersection product in the context of tangential de Rham cohomology and homology of tangential currents. We apply the formula to get a dynamical Lefschetz formula for …