The paper introduces geodesic φ-convex functions and their properties.
problem Generalizing geodesic functions to φ-convex functions.
method Introducing geodesic φ-convex functions and investigating their properties.
result Characterization of geodesic φ-convex functions via their φ-epigraphs.
Geodesic convex optimization extends convex optimization to manifolds.
problem Optimizing non-convex functions on manifolds.
method Introducing geodesic convexity on manifolds.
result Certain non-convex problems can be formulated as geodesically convex optimization problems.
Self-crossing geodesics on convex surfaces are studied.
problem Understanding patterns of geodesics crossing themselves.
method Analyzing closed geodesics on convex surfaces.
result Self-crossing geodesics exist on convex surfaces.
Upper bound found for geodesic curvature on convex surfaces.
problem Bounding the total curvature of geodesics on convex surfaces.
method Provided a universal upper limit for minimizing geodesics.
result Established a universal upper bound for total curvature.
Long geodesics imply a special shape of convex bodies.
problem Understanding the geometry of convex surfaces.
method Intrinsic geometry of convex surfaces and proof by contradiction.
result Long geodesics on a convex surface imply the shape is an isosceles tetrahedron.
The paper proves geodesic connectedness for convex functions in space-times.
problem Geodesic connectedness of space-times and semi-Riemannian manifolds.
method Geometric-topological proofs for specific classes of space-times.
result Geodesic connectedness for null-disprisoning space-times and timelike strictly convex hypersurfaces.
New global section found for geodesic flows on convex hypersurfaces.
problem Finding global sections for geodesic flows on convex hypersurfaces.
method Constructing a global hypersurface of section with an isometric involution.
result Generalized Birkhoff annulus to higher dimensions.
Established strong geodesic convex functions and their properties.
problem Geodesic convex functions and monotone vector fields on Riemannian manifolds.
method Characterization and relation establishment for strong geodesic convex functions.
result Relation between variational inequality solutions and strict minimizers for multiobjective programming.
Geodesic flow mixing on convex projective manifolds proven.
problem Understanding mixing properties of geodesic flow on convex projective manifolds.
method Introduced biproximal unit tangent bundle and proved mixing properties.
result Geodesic flow is topologically mixing on biproximal unit tangent bundle.
Geodesic convexity types differ in Riemannian manifolds.
problem Characterizing geodesic convexity types in Riemannian manifolds.
method Reverse engineering to characterize manifolds with coinciding convexity types.
result Characterized complete manifolds with coinciding geodesic convexity types.
Stable nets on convex hypersurfaces maintain their shape under small perturbations.
problem Maintaining the shape of nets on convex surfaces under slight changes.
method Constructing stable geodesic nets on convex hypersurfaces.
result Stable geodesic nets on convex hypersurfaces do not change shape under small perturbations.
Study on Mabuchi functional's convexity using ε-geodesics.
problem Understanding the convexity of the Mabuchi functional.
method Analysis of ε-geodesics to study the Mabuchi functional's convexity.
result Uniform fiberwise non-degeneracy of geodesics when Mabuchi functional is ε-affine.
Extends DCP framework to Hadamard manifolds for geodesically convex functions.
problem Verifying convexity in nonlinear programs on Hadamard manifolds.
method Introduces Disciplined Geodesically Convex Programming (DGCP) framework, defining compositions and transformations for geodesically convex functions.
result Allows verification of geodesic convexity for a broader range of functions, including statistical estimators and matrix-valued optimization.
The study extends flat plane embedding results to spaces with convex geodesic bicombings.
problem Embedding flat planes in spaces with non-unique geodesics.
method Convexity assumptions on distance functions along geodesics.
result Results on flat plane embedding remain valid in spaces with convex geodesic bicombings.
Proves stability of convex spheres with similar geodesic lengths.
problem Stability of convex spheres with specific geodesic properties.
method Proves C^0 Cheeger-Gromov closeness to the round sphere.
result Strictly convex 2-spheres are close to the round sphere.
The paper proves strict convexity of the Mabuchi functional for geodesics connecting energy minimizers.
problem Proving strict convexity of the Mabuchi functional for geodesics.
method Explicit formula for the complex Hessian of the weighted log-Bergman kernel, and proof by showing geodesics must be non-degenerate and smooth.
result Strict convexity of the Mabuchi functional along geodesics connecting energy minimizers.
The paper generalizes λ-radial contraction and introduces pλ-convex sets in Riemannian manifolds.
problem Generalizing λ-radial contraction and defining pλ-convex sets in Riemannian manifolds. method Developed the concept of pλ-convex function and provided counterexamples and relations between geodesic convex sets and pλ-convex sets. result Under certain conditions, geodesic convex sets and pλ-convex sets are equivalent. Entropy study of geodesic flow on convex projective surfaces.
problem Entropy of Sinai-Ruelle-Bowen measure on convex projective surfaces.
method Analysis of Hilbert area and Blaschke metric.
result Entropy tends to zero if and only if the Hilbert area tends to infinity.
First-order methods tackle g-convex optimization on Hadamard manifolds.
problem Geodesically convex optimization on nonlinear metric spaces.
method Iteration complexity analysis for first-order algorithms.
result Upper bounds for global complexity of g-convex optimization.
The study finds at least two short, simple geodesic chords on a disk with convex boundary.
problem Existence of short, simple geodesic chords on a 2-disk with convex boundary.
method Proof of existence using Riemannian geometry and bounds on lengths.
result Existence of at least two short, simple orthogonal geodesic chords on a 2-disk with convex boundary.
Asymptotic geodesics in convex polygons are convex for large distances.
problem Understanding convexity of geodesics in Hilbert geometry.
method Analyzing the distance function between asymptotic geodesics for large t.
result The distance function between asymptotic geodesics is convex for sufficiently large t.
Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.
problem Characterizing totally geodesic submanifolds in geometrically finite manifolds.
method Analysis of totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds.
result Every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur.
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
Study convexity of geodesics and balls in Outer space.
problem Convexity properties of geodesics and balls in Outer space.
method Introduced balanced folding paths and used them to show weak convexity of out-going balls.
result Weak convexity of out-going balls in Outer space.
Geodesic flows on specific manifolds are structurally stable.
problem Stability of geodesic flows on compact manifolds without conjugate points.
method Analyzing the C∞ compact manifold (M,g) with quasi-convex universal covering and divergent geodesic rays. result Proved the C1-stability conjecture for geodesic flows of compact manifolds. New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applicat…
The paper proves a geodesic sandwich theorem and applies it to manifold inequalities.
problem Proving inequalities in Riemannian manifolds with bounded curvature.
method Geodesic convex functions and sectional curvature bounds.
result Gradient of convex functions is orthogonal to geodesics.
The paper proves the existence and properties of geodesics on convex surfaces.
problem Existence and properties of geodesics on convex surfaces with free boundaries.
method Free boundary curve shortening flow on closed surfaces with strictly convex boundary.
result Existence of two free boundary embedded geodesics and geodesics with Morse Index 1 and 2.
Geodesically convex functions are continuous on Riemannian manifolds.
problem Continuity of geodesically convex functions on Riemannian manifolds.
method Proof of continuity using geodesic convexity and addressing a gap in existing proof.
result All geodesically convex functions are continuous in the interior of their domain on Riemannian manifolds.
The paper proves properties of geodesic planes in a specific type of 3-manifold.
problem Characterizing geodesic planes in a specific class of 3-manifolds.
method Analyzing geodesic planes within the convex core of a hyperbolic 3-manifold.
result Geodesic planes are either closed or dense, with only countably many being closed.
We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofa…
New constructions show stable geodesics and figure-eights in convex hypersurfaces.
problem Constructing stable geodesics and figure-eights in convex hypersurfaces.
method Explicit billiard trajectories with controlled parallel transport in convex polytopes.
result Construction of stable figure-eights and index-zero geodesics in convex hypersurfaces.
In non-compact manifolds, geodesic flowers exist.
problem Existence of geodesic flowers in non-compact manifolds.
method Proving the existence of non-trivial geodesic flowers in complete non-compact manifolds with locally convex ends.
result Non-trivial geodesic flowers exist in every complete non-compact manifold with locally convex ends.
Study pinches eigenvalues and curvatures of convex hypersurfaces to make them nearly geodesic spheres.
problem Pinching eigenvalues and curvatures of convex hypersurfaces to understand their geometric properties.
method Pinching Heintze-Reilly's inequality via sectional curvature upper bounds.
result Closed convex hypersurfaces are Hausdorff close and almost isometric to geodesic spheres.
Geodesic tomography identifies piecewise constants on convex manifolds.
problem Determining piecewise constant functions on nontrapping manifolds.
method Iterating local uniqueness results based on geodesic integrals.
result Piecewise constant functions are uniquely determined by their geodesic integrals.
Billiard trajectories and geodesics are closely related geometrically.
problem Understanding the relationship between billiard trajectories and geodesics on surfaces.
method Establishing mutual approximation results for billiard trajectories and geodesic segments on surfaces.
result For Riemannian billiard tables, there are families of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory.
Study geodesic distances and convexity in contact sets.
problem Understanding geodesic distances and convexity in contact sets.
method Extending results on quasi-psh functions and big cohomology classes, studying Monge-Ampère measures on contact sets.
result Convexity of the K-energy in big and nef cohomology classes.
The paper surveys partial results on convex real projective orbifolds with specific ends.
problem Characterizing convex real projective orbifolds with radial or totally geodesic ends.
method Proving homeomorphisms between deformation spaces and strata of projective structures.
result A homeomorphism between deformation spaces of convex real projective structures on orbifolds with radial or totally geodesic ends.
New study shows acceleration in hyperbolic spaces is impossible for strongly geodesically convex functions.
problem Acceleration in hyperbolic spaces for strongly geodesically convex functions is impossible.
method Perturbing hard functions with sums of bump functions chosen by a resisting oracle.
result Acceleration is unachievable for any deterministic algorithm in hyperbolic spaces for strongly geodesically convex functions.
The paper studies conjugate loci on convex surfaces and proves a minimum of four cusps.
problem Understanding conjugate loci on convex surfaces.
method Analyzing geodesics and their envelopes on convex surfaces.
result The conjugate locus of a generic point on a convex surface must have at least four cusps.
Establishes geodesic stability for Kähler metrics, proving existence of constant scalar curvature.
problem Existence of constant scalar curvature Kähler metrics.
method Exploring metric geometry of Mabuchi geodesic rays and uniform convexity properties of Kähler metrics space.
result Essentially optimal form of Donaldson's geodesic stability conjecture proved.
Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.
problem Volume preserving Gauss curvature flow of convex hypersurfaces in hyperbolic space.
method Volume preserving flow with speed given by Gauss curvature power α, using Alexandrov reflection and hyperbolic curvature measures.
result Smooth solution remains convex and converges to a geodesic sphere exponentially.
Every convex set in a generic Riemannian manifold has peculiar properties.
problem Characterizing convex sets in Riemannian manifolds.
method Analyzing geodesics and hypersurfaces in Riemannian manifolds.
result Convex sets in generic Riemannian manifolds are strictly convex if bounded by smooth hypersurfaces.
Geodesic balls are isoperimetric in hyperbolic spaces with certain densities.
problem Proving isoperimetric properties in hyperbolic spaces with specific densities.
method Using geodesic balls and radial, strictly log-convex densities.
result Geodesic balls are isoperimetric in real hyperbolic space HRn. Paper analyzes convergence of proximal algorithm in metric spaces without geodesic convexity.
problem Analyzing convergence of proximal algorithm in general metric spaces.
method Analysis of the Wasserstein proximal algorithm without geodesic convexity assumption.
result Establishes unbiased and linear convergence rate for proximal algorithm under natural Wasserstein inequality.
Study finds geodesic networks for surfaces with convex boundary.
problem Finding geodesic networks for surfaces with convex boundary.
method Investigates free boundary geodesic networks in surfaces with non-negative sectional curvature and convex boundary.
result Existence of a geodesic network realizing the first width of a surface with non-negative sectional curvature and strictly convex boundary.
Study coning totally geodesic boundaries of hyperbolic manifolds.
problem Understanding metrics on coned-off spaces of hyperbolic manifolds.
method Analyzing the geometric and group-theoretic properties of coned-off spaces.
result Explicit conditions for negatively curved metrics and locally convex subsets.
Study geodesics on flat tori, focusing on convex bodies.
problem Analyze geodesics orthogonal to convex subsets on flat tori.
method Define anisotropic Sobolev spaces and study properties of geodesics.
result Compute residues of geometric Epstein function in terms of intrinsic volumes.