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.
Free finite group actions on non-positively curved 3-manifolds
problem Resolution of the remaining case of graph manifolds
method Proving the existence of a G-invariant NPC metric result Resolution of the remaining case of graph manifolds
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.
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.
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.
We present short proofs of all known topological properties of general Busemann G-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally G-homogeneous Busemann G-spaces are homeomorphic and strongly topologically homogeneous. This is a key r…
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.
NPC tackles class imbalance in datasets by progressively considering more neighbors.
problem Class imbalance in datasets causes poor performance in classification tasks.
method Inspired by k-NN, NPC progressively considers more neighbors and uses local-global grading.
result NPC outperforms five representative algorithms on fifteen imbalanced datasets.
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.
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.
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.
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.
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.
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…
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.
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.
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 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.
Biography of mathematician Herbert Busemann.
problem N/A
method N/A
result N/A
The study proves local and global Holder continuity of quasi-n-harmonic mappings into spaces with non-positive curvature.
problem Holder continuity of quasi-n-harmonic mappings into metric spaces with non-positive curvature.
method Local and global Holder continuity proved using Euclidean domains and bounded Lipschitz domains.
result Local and global Holder continuity of quasi-n-harmonic mappings.
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.
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 …
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.
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.
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.
New convex body defined to prove Busemann Random Simplex Inequality.
problem Proving Busemann Random Simplex Inequality for general p.
method Defining NpL and proving isoperimetric inequality for its dual. result Equivalent to Lp-Busemann-Petty centroid inequality for p=1,2. Study extends convexity in curved spaces using fractional integrals.
problem Extending convexity to curved spaces with nonpositive curvature.
method Introducing (geodesically) h-convex functions and using Katugampola's fractional integrals. result Essentially sharp estimate involving squared distance mappings.
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.
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.
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…
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.
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.
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…
We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every n-dimensional homogeneous ANR is a topological n-manifold, whereas the Busemann Conjecture asserts that every n-dimensional G-space is a topological n-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 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.
This article generalizes the work of Ballmann and Światkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.
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.
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.
Survey on harmonic maps in non-smooth spaces, focusing on rigidity.
problem Rigidity phenomena in non-smooth spaces.
method Regularity theory of harmonic maps to non-smooth targets.
result Generalizations of Margulis superrigidity and holomorphic rigidity of Teichmüller space.
In this note we prove that any closed graph manifold admitting a metric of non-positive sectional curvature (NPC-metric) has a finite cover, which is fibered over the circle. An explicit criterion to have a finite cover, which is fibered over the circle, is presented for the graph manifolds of certain class.
A test measures artificial agents' human-like behavior in video games.
problem Measuring the believability of artificial agents' human-like behavior.
method Developed a non-parametric two-sample hypothesis test.
result The p-value correlates with human judgment of human-like behavior. 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.
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.
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.
Optimal geodesics connect boundary points in Teichmüller space.
problem Finding optimal geodesics between boundary points of Teichmüller space.
method Analyzing horofunctions and Teichmüller geodesics.
result There is a unique optimal geodesic connecting boundary points.
The study of harmonic maps into non-positively curved spaces proves a Bochner formula.
problem Understanding harmonic maps in non-positively curved spaces.
method Expanding domain variation and Bochner formulas in terms of curvature.
result Harmonic maps from spaces of non-negative Ricci curvature into non-positively curved spaces have subharmonic energy density.