Biography of mathematician Herbert Busemann.
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48 results for “Busemann”
Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
Busemann and MCPmath.DG
Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.
problem Understanding the structure of Busemann spaces with measures.
method Analyzing geodesic completeness and non-collapse assumptions.
result Rigidity and structure theorems for Busemann spaces with MCP.
Decomposes Busemann spaces into simpler structures.
problem Understanding the structure of Busemann spaces.
method Topological decomposition and isometry analysis.
result Busemann spaces can be decomposed into products of simpler spaces.
We present short proofs of all known topological properties of general Busemann -spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally -homogeneous Busemann -spaces are homeomorphic and strongly topologically homogeneous. This is a key r…
First example of open manifold with positive Ricci curvature and non-proper Busemann function.
problem Counterexample to Busemann function properness in open manifolds with nonnegative Ricci curvature.
method Provided an open manifold with positive Ricci curvature and non-proper Busemann function.
result First example of open manifold with positive Ricci curvature and non-proper Busemann function.
Study calculates Hessian of Busemann function on Damek-Ricci spaces.
problem Calculating Hessian of Busemann function on Damek-Ricci spaces.
method Calculates eigenvalues of Hessian and proves positive definiteness.
result Hessian of Busemann function is positive definite.
The Hessian of Busemann functions is positive definite on certain Hadamard manifolds.
problem Analyzing the Hessian of Busemann functions on Hadamard manifolds.
method Investigating the Hessian of Busemann functions on harmonic Damek-Ricci spaces and Hadamard manifolds.
result The Hessian of Busemann functions is positive definite on certain Hadamard manifolds.
New convex body defined to prove Busemann Random Simplex Inequality.
problem Proving Busemann Random Simplex Inequality for general p.
method Defining and proving isoperimetric inequality for its dual.
result Equivalent to -Busemann-Petty centroid inequality for p=1,2.
Busemann G-spaces with Finsler metrics
problem Admitting a differentiable DC atlas with a continuous Finsler metric
method Comparison geometry
result Generalizes previous work on G-spaces with Riemannian curvature bounds
Busemann points are sparse in Teichmüller spaces.
problem Characterizing the limits of geodesic rays in Teichmüller spaces.
method Proof of nowhere density of Busemann points in horoboundaries.
result Teichmüller metrics lack non-positive curvature.
Commentary on Busemann's foundational geometry papers.
problem Foundational axiomatic geometry and Minkowski spaces.
method Translation and analysis of three seminal papers.
result Early ideas and techniques for Minkowski spaces.
Complex and Quaternionic Analogues of Busemann's Random Simplex and Intersection Inequalitiesmath.MG
The paper extends Busemann's inequalities to complex and quaternionic spaces.
problem Extending Busemann's inequalities to complex and quaternionic vector spaces.
method Proof leverages a monotonicity property under symmetrization with respect to complex or quaternionic hyperplanes.
result Standard Steiner symmetrization does not exhibit the monotonicity property in complex or quaternionic spaces.
Abstract manifolds split with no conjugate points using special functions.
problem Splitting compact manifolds without conjugate points.
method Developed functions that vanish under certain conditions on central Busemann functions.
result Central Busemann functions split linearly under specific conditions.
We determine the asymptotic behaviour of extremal length along arbitrary Teichmüller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichmüller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the…
The study proves properties of intersections of horospheres in harmonic spaces.
problem Properties of intersections of horospheres in harmonic spaces.
method Constructing volume preserving mappings using Busemann functions.
result Upper bound of the volume of intersection of horospheres is independent of Busemann function differences.
Study of rays and co-rays in Wasserstein space, introducing Busemann functions.
problem Understanding rays and co-rays in Wasserstein space.
method Representation of rays as probability measures, existence of co-rays, introduction of Busemann functions.
result Existence and properties of co-rays and Busemann functions.
We discuss two different in general natural approaches to the ideal closure and ideal boundary of Busemann nonpositively curved metric space. It is shown that the identity map of the space admits surjective continuation from its coarse ideal closure to the weak one. We consider some situations when these closures coinc…
Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicingcs.LG
Researchers find Busemann functions in Wasserstein space, enabling efficient projections and distances.
problem Defining Busemann functions in Wasserstein space for efficient data projections and distances.
method Investigated existence and computation of Busemann functions in Wasserstein space, establishing closed-form expressions for specific cases.
result Explicit projection schemes for probability distributions on \(\mathbb{R}\) enable novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets.
The paper explores isometric models and Busemann functions for Funk and Hilbert discs.
problem Exploring isometric models and Busemann functions for Funk and Hilbert discs.
method Finding and describing isometric models and computing Busemann functions.
result Proving asymptotic harmonicity of the Funk disc and showing its dependence on measure.
In this paper, we shall show that the metric boundary of the Teichmueller space with respect to the Teichmueller distance contains non-Busemann points when the complex dimension of the Teichmueller space is at least two.
The paper explores curvature conditions on metric spaces and their properties.
problem Understanding curvature in metric spaces and its implications.
method Introducing Busemann concavity, applying comparison geometry, and using uniform smoothness.
result Spaces with certain curvature bounds have specific geometric properties.
New optimal isosystolic inequality found for Finsler reversible 2-tori.
problem Optimal isosystolic inequalities on Finsler tori.
method Survey and new inequality derived from prior work.
result Busemann-Hausdorff area of a Finsler reversible 2-torus with unit systole is at least π/4.
A new depth measure and median defined on Hadamard manifolds.
problem Statistical depth and median on Hadamard manifolds.
method Horospherical depth and Busemann median defined using renormalized distance functions.
result The Busemann median exists for every Borel probability measure on Hadamard manifolds.
No non-trivial convex functions on certain Finsler manifolds.
problem Existence of convex functions on Finsler manifolds.
method Analyzing Holmes- Thompson volume and Busemann function properties.
result Non-compact Finsler manifolds with infinite Holmes- Thompson volume have no non-trivial convex functions.
We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is …
Paper solves isoperimetric problem in specific Finsler space.
problem Isoperimetric problem in 2D Finsler space with k=0.
method Using Busemann-Hausdorff area, proved circle maximizes area.
result Circle centered at origin maximizes area in isoperimetric problem.
Researchers create metrics on hyperbolic space's tangent bundle.
problem Constructing metrics on the unit tangent bundle of hyperbolic space.
method Using Hopf coordinates and Busemann functions, they constructed a flow-invariant metric.
result The unit tangent bundle of hyperbolic space is a homogeneous space under specific groups.
On Hilbert's fourth problemmath.HO
Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. Th…
Paper studies metric properties on geodesic spaces to characterize curvature negativity.
problem Characterizing curvature negativity in geodesic spaces.
method Introduces metric properties via metric projection and applies to Alexandrov and Busemann NPC spaces.
result Proves both properties characterize non-positivity of sectional curvature on Riemannian manifolds.
We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every -dimensional homogeneous ANR is a topological -manifold, whereas the Busemann Conjecture asserts that every -dimensional -space is a topological -manifold. The key object in bo…
A new learning method using hyperbolic geometry for class labels.
problem Class label representation and learning in machine learning.
method Hyperbolic Prototype Learning with a new loss function based on hyperbolic geometry.
result Hyperbolic Prototype Learning is equivalent to logistic regression in the one-dimensional case.
Study weakly weighted Einstein-Finsler metrics, showing specific curvature properties and characterizing them.
problem Characterizing weakly weighted Einstein-Finsler metrics.
method Showed isotropic S-curvature under certain conditions. Characterized via navigation expressions and and .
result Weakly weighted Einstein-Kropina metrics have isotropic S-curvature and can be completely characterized.
Paper extends tree bijection for hyperbolic surfaces without requiring cusps.
problem Computing volumes and distances on hyperbolic surfaces without cusps.
method Extend tree bijection to half-tight cylinders, using Busemann function.
result Tree bijection can now be applied to surfaces without cusps.
The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.
problem Characterizing the horoboundary of Teichmüller space.
method Using the relationship between the horofunction and visual compactifications of Teichmüller spaces.
result The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.
Recently, the old notion of causal boundary for a spacetime V has been redefined in a consistent way. The computation of this boundary for a standard conformally stationary spacetime V = R x M, suggests a natural compactification associated to any Riemannian metric on M or, more generally, to any Fin…
Subdifferential theory and the Fenchel conjugate via Busemann functions on Hadamard manifoldsmath.DG
The paper defines subdifferentials on Hadamard manifolds and identifies conditions for Fenchel conjugate equality.
problem Understanding convex analysis on Riemannian manifolds.
method Using Busemann functions to define subdifferentials and investigate Fenchel conjugate equality.
result Identifies conditions for equality in the Fenchel-Young inequality on Hadamard manifolds.
The study characterizes Finsler metrics and proves their rigidity.
problem Characterizing and proving rigidity of Busemann convex Finsler metrics.
method Proving Finsler metrics are nonpositively curved if and only if they are affinely equivalent to a Riemannian metric of nonpositive sectional curvature.
result Finsler metrics are precisely Berwald metrics of nonpositive flag curvature.
We show that the co-rays to a ray in a complete non-compact Finsler manifold contain geodesic segments to upper level sets of Busemann functions. Moreover, we characterise the co-point set to a ray as the cut locus of such level sets. The structure theorem of the co-point set on a surface, namely that is a local tree, …
Extends topological results for nonpositive curvature spaces.
problem Characterize and understand topological properties of BNPC spaces.
method Extends results from CAT(0) spaces to BNPC spaces, using links and geodesic bicombings.
result Locally BNPC topological manifolds have discrete singular sets and are homeomorphic to Euclidean space.
We investigate the finiteness structure of a complete non-compact -dimensional Riemannian manifold whose radial curvature at a base point of is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than . We show, as our main theorem, that all Buse…
The paper extends metric space concepts from linear to non-linear settings.
problem Extension of linear concepts to metric spaces.
method Extension of symmetric orthogonality and non-expansive projections from linear to metric spaces, considering spaces with non-positive curvature.
result Equivalence of symmetric orthogonality and non-expansive projections in spaces with non-positive curvature.
New theorem splits weighted Lorentz-Finsler manifolds into simpler parts.
problem Understanding the geometry of weighted Lorentz-Finsler manifolds.
method Developed a splitting theorem using weighted Berwald spacetimes and Busemann functions.
result Weighted Lorentz-Finsler manifolds with certain properties split into simpler isometric translations.
Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann--Petty centroid inequality.
Study noncompact RCD(0,N) spaces with linear volume growth, proving diameter bounds and a splitting theorem.
problem Understanding non-compact RCD(0, N) spaces with linear volume growth.
method Analyzing properties of level sets and applying geometric inequalities.
result Diameter of level sets of a Busemann function grows at most linearly.
In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, -flow, for Here we investigate the asymptotic behavior of the planar -flow for in the class of smooth, origin-symme…
Graphical theory of binary Hamiltonian forms and their values.
problem Understanding values of binary Hamiltonian forms.
method Graphical theory, maximal orders, Busemann distances, equivariant cellulation.
result Combinatorial description of binary Hamiltonian form values.
We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen's S-curvature vanishes everywhere. Moreover, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.