Paper examines non-positive curvature properties of Hilbert metric in convex domains.
problem Investigating curvature properties of Hilbert metric in convex domains.
method Surveying and proving relationships among concepts, showing conditions for rigidity.
result If Hilbert metric is Berwald, domain is an ellipsoid and metric is Riemannian.
The Hilbert metric on Teichmüller space is studied for punctured surfaces.
problem Understanding the geometry of Teichmüller space.
method Parametrization of Teichmüller space and study of Hilbert metric.
result Every earthquake ray is an almost geodesic under the Hilbert metric.
Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
Three types of Einstein metrics are disqualified as potential local maxima.
problem Identifying local maxima of the Hilbert action in Einstein metrics.
method Analysis of three infinite families of neutrally stable homogeneous Einstein metrics.
result Three families of Einstein metrics ruled out as local maxima.
Theory developed for Hilbert geometry over valued fields, linking real and non-Archimedean geometries.
problem Understanding Hilbert geometry over general valued fields and their limits.
method Developed a theory of Hilbert geometry over general ordered valued fields, proving ultralimit results.
result Ultralimit of rescaled real Hilbert geometries is isometric to a non-Archimedean Hilbert metric space.
New Hilbert bundles with ends defined from indexed bases.
problem Defining new structures in Hilbert bundles.
method Indexed bases and unitary operators of finite propagation.
result Characteristic classes of Hilbert bundles with ends.
We survey some basic geometric properties of the Funk metric of a convex set in Rn. In particular, we study its geodesics, its topology, its metric balls, its convexity properties, its perpendicularity theory and its isometries. The Hilbert metric is a symmetrization of the Funk metric, and we show some pro…
David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain Ω in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of the boundary of Ω and refers to a theorem by Busemann and Mayer that…
Infinite-dimensional geometry: completeness and geodesics in Hilbert manifolds.
problem Failure of Hopf--Rinow theorem in Hilbert manifolds.
method Investigates conformal flexibility and completeness properties in infinite-dimensional settings.
result Conformal class of metrics on Hilbert manifolds contains complete representatives.
This study describes the Fisher-Rao metric on Gaussian measures in infinite-dimensional spaces.
problem Understanding the Fisher-Rao metric in infinite-dimensional Gaussian settings.
method Explicit description and generalization of finite-dimensional quantities to infinite-dimensional Hilbert spaces.
result The Fisher-Rao metric and related geometric quantities generalize from finite to infinite dimensions.
Study of Einstein-Hilbert actions with non-symmetric metrics and torsion.
problem Exploring the effects of torsion on Einstein-Hilbert actions.
method Developed general formulae for pressure and density, derived energy-momentum tensor, and generalized Bianchi type-I model.
result Obtained expressions for pressure and density with non-symmetric metrics.
The paper characterizes hyperbolic spaces using metrics' entropy.
problem Characterizing hyperbolic spaces using metrics' entropy.
method Using Poincaré exponent and volume growth entropy.
result Generalizes results in Hilbert and Riemannian metrics.
We detect Hilbert manifolds among isometrically homogeneous metric spaces and apply the obtained results to recognizing Hilbert manifolds among homogeneous spaces of the form G/H where G is a metrizable topological group and H is a closed balanced subgroup of G.
Study shows non-positivity of Einstein-Hilbert action for certain metrics.
problem Analyzing the non-positivity of the Einstein-Hilbert action for specific metrics.
method Using spectral triples and modular operator computations.
result Recovery of earlier results on noncommutative tori and new Gauss-Bonnet theorem.
The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…
New metric defined for bounded symmetric domains.
problem Defining a new metric for bounded symmetric domains.
method Using generalized Hilbert metric and Borel embedding.
result The new metric differs from Carathéodory and Bergman metrics except for complex hyperbolic space.
We give first examples of finitely generated groups having an intermediate, with values in (0,1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert …
Embeds persistence diagrams into Hilbert spaces to use kernel methods.
problem No inner product structure on persistence diagrams.
method Shows non-embeddability of persistence diagrams into Hilbert spaces.
result Persistence diagrams with bottleneck distance do not coarse embed into Hilbert spaces.
A strictly convex real projective orbifold is equipped with a natural Finsler metric called the Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that the marked Hilbert length spectrum determines the projective structure only up to …
Study on embedding persistence diagrams into Hilbert spaces, focusing on metric distortion.
problem Understanding metric properties of persistence diagrams in Hilbert spaces.
method Investigate embedding persistence diagrams into separable Hilbert spaces using bi-Lipschitz maps.
result Impossible to find a bi-Lipschitz embedding into finite-dimensional Hilbert spaces.
Study Hilbert's projective metric for bounded growth functions leading to Sinkhorn's algorithm convergence.
problem Optimal transport in unbounded settings with heavy-tailed distributions.
method Hilbert's projective metric for integrable functions of bounded growth, kernel integral operators as contractions.
result Exponential convergence of Sinkhorn's algorithm for light-tailed marginal distributions.
It is shown that the Hilbert geometry (D,hD) associated to a bounded convex domain D⊂En is isometric to a normed vector space (V,∣∣⋅∣∣) if and only if D is an open n-simplex. One further result on the asymptotic geometry of Hilbert's metric is obtained with corollaries for the behavior …
We show that the spheres in Hilbert geometry have the same volume growth entropy as those in the Lobachevsky space. We give the asymptotic estimates for the ratio of the volume of metric ball to the area of the metric sphere in Hilbert geometry. Derived estimates agree with the well-known fact in the Lobachevsky space
Study on spaces of nonnegatively curved surfaces and their topological properties.
problem Characterizing spaces of nonnegatively curved metrics on surfaces.
method Analysis of homeomorphism types of spaces of smooth complete nonnegatively curved metrics on surfaces.
result Spaces of metrics on surfaces have specific topological properties, including being homeomorphic to Hilbert spaces or countable powers of Hilbert cubes.
We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov's notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum-Connes conjecture holds for metric …
Optimal inequalities found between Riemannian and Hilbert metrics in convex projective domains.
problem Finding optimal bounds between Riemannian and Hilbert metrics in convex projective domains.
method Optimal control techniques applied to Riemannian metrics induced by centro-affine hypersurface immersions.
result Optimal inequalities between Riemannian and Hilbert metrics for a class of convex projective domains.
We show that every complete metric space is homeomorphic to the precise locus of zeros of an entire analytic map from a Hilbert space to a Banach space. As a corollary, every complete separable metric space is homeomorphic to the precise locus of zeros of an entire analytic map between two separable complex Hilbert spa…
The paper proves inequalities and consequences in Hilbert metrics and Hitchin representations.
problem Volume entropy rigidity and length spectrum comparison in specific geometric settings.
method Sharp distance inequalities and volume growth analysis.
result Volume entropy rigidity for Hilbert geometries and length spectrum comparison for Hitchin representations.
Constructs stable Hilbert bundles on curves using Diophantine approximation.
problem Constructing stable Hilbert bundles on complex projective curves.
method Investigating arithmetic properties of the upper half plane and applying Diophantine approximation to bound Hermitian-Einstein metrics.
result Constructs Hilbert bundles with Hermitian-Einstein metrics on curves of positive genus.
We consider Hilbert and Funk geometries on a strongly convex domain in the Euclidean space. We show that, with respect to the Lebesgue measure on the domain, Hilbert (resp. Funk) metric has the bounded (resp. constant negative) weighted Ricci curvature. As one of corollaries, these metric measure spaces satisfy the cur…
We show that the volume entropy of the Hilbert metric on a closed convex projective surface tends to zero as the corresponding Pick differential tends to infinity. The proof is based on the theorem, due to Benoist and Hulin, that the Hilbert metric and Blaschke metric are comparable.
Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.
problem Stability of critical points of the generalized Einstein Hilbert action in non-Kähler Calabi-Yau theory.
method Analysis of Bismut Hermitian Einstein manifolds and Bismut flat pluriclosed steady solitons, proving stability conditions.
result All Bismut Hermitian Einstein manifolds are linearly stable, and all Bismut flat pluriclosed steady solitons with positive Ricci curvature are linearly strictly stable.
Solves a Dirichlet problem for flat metrics on Riemann surfaces with boundary.
problem Solving a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary.
method Proves solvability using flat hermitian metrics and factorization results.
result Solves the Dirichlet problem for flat metrics on Riemann surfaces with boundary.
Improves Kalman filter convergence analysis using Hilbert metric.
problem Improving Kalman filter convergence over general graphical models.
method Analyzes contraction of Riccati map in probability distributions with Hilbert metric.
result Improves understanding of filtering maps in probability spaces.
Sharp estimates for Finsler metrics in convex domains.
problem Estimating distances in Finsler metrics near convex points.
method Sharp estimates for intrinsic distances of Finsler metrics.
result Characterization of k-quasi hyperbolic metric in convex geometry. Vanishing geodesic distances in infinite dimensions can be created.
problem Vanishing geodesic distances in infinite-dimensional spaces.
method Constructing a weak Riemannian metric in a Hilbert manifold.
result Vanishing geodesic distances can be engineered.
A new method estimates multi-dimensional value distributions using Hilbert space embeddings.
problem Estimating value distributions in complex, multi-dimensional reinforcement learning settings.
method Hilbert space mappings and kernel mean embeddings to estimate the kernel mean embedding of multi-dimensional value distributions.
result Uniform convergence guarantees and robust off-policy evaluation demonstrated in simulations.
Timelike geometry of spherical simplices is shown to be isometric to vector spaces.
problem Characterizing the geometry of spherical simplices.
method Proved isometry to vector spaces with a timelike norm.
result Timelike spherical Hilbert geometry of simplices is isometric to a union of six copies of vector spaces.
Study of Einstein-Hilbert action on metric-affine spaces with connections.
problem Formulating and solving variational problems for mixed Einstein-Hilbert action.
method Developed variational formulas for extrinsic geometry, derived Euler-Lagrange equations, and characterized critical points.
result Derived new equations analogous to Einstein and Cartan equations, with a new Ricci type tensor.
Paper develops metrics for random dynamical systems using vector-valued RKHSs.
problem Creating metrics for random nonlinear dynamical systems.
method Develops metrics on random dynamical systems using Perron-Frobenius operators in vector-valued reproducing kernel Hilbert spaces (vvRKHSs). Uses operator-valued kernels and time-wise independence criteria.
result Extends existing metrics for deterministic systems and introduces kernel maximal mean discrepancy for random processes.
New hyperkähler metrics constructed via Riemann-Hilbert problems.
problem Constructing hyperkähler metrics on complex integrable systems with singular fibers.
method Solving Riemann-Hilbert problems and isomonodromic deformations to obtain Darboux coordinates.
result Holomorphic symplectic form constructed, leading to hyperkähler metric.
New stable metric found on a complex space.
problem Stability of a non-symmetric metric on a complex space.
method Proving stability with respect to the Einstein-Hilbert action.
result First known example of a non-symmetric metric of positive scalar curvature.
Study of Hilbert schemes and Coulomb branches of hypertoric varieties.
problem Understanding the geometry and topology of Coulomb branches of hypertoric varieties.
method Investigation of transverse equivariant Hilbert schemes and Hamiltonian reductions, proposing new metrics.
result Coulomb branches of hypertoric varieties can be constructed as Hilbert schemes or Hamiltonian reductions.
New formula for Lichnerowicz Laplacian on homogeneous spaces.
problem Finding new Einstein metrics on homogeneous spaces.
method Using Casimir operators to derive a new formula for the Lichnerowicz Laplacian.
result Derives many new Einstein metrics stable in the Einstein-Hilbert sense.
The paper describes all isometry-invariant Finsler metrics on Hilbert spaces.
problem Analyzing Finsler metrics invariant under isometries.
method Analytic description and characterization of all isometry-invariant Finsler metrics.
result The only possible linear maps under which the metric is invariant are scalar multiples of isometries.
The paper characterizes arithmetic metrics in coarsely geometric settings.
problem Characterizing arithmetic metrics in coarsely geometric settings.
method Using coarse-geometric commensurators and under the Hilbert-Smith conjecture.
result Positive answer in general and unconditional for specific cases.
It is shown that the Hilbert metric on the interior of a convex polytope is bilipschitz to a normed vector space of the same dimension.
This paper solves Hilbert's fourth problem for constant curvature metrics.
problem Classifying metric geometries with shortest straight lines in constant curvature settings.
method Analyzing Finsler manifolds with constant flag curvature, deriving distance formulas, and proving global geometry theorems.
result Complete characterization of global geometry for constant flag curvature metrics.