The paper proves geometric inequalities and their stabilities for curves in hyperbolic space.
problem Geometric inequalities and their stabilities for curves in hyperbolic space.
method Curve flow for shifted principal curvatures, Heintze-Karcher type inequality for h-convex curves.
result Geometric inequalities and their stabilities for curves in hyperbolic space.
New method optimizes on curved manifolds without curvature dependence.
problem Curvature-dependent regret in online optimization on Hadamard manifolds.
method Riemannian online gradient descent for h-convex functions.
result Established O ( T ) O(\sqrt{T}) O ( T ) and O ( log ( T ) ) O(\log(T)) O ( log ( T )) regret guarantees, curvature-independent. The flow preserves curvature and converges to a geodesic sphere in hyperbolic space.
problem Preserving curvature in hyperbolic space.
method Flow of hypersurfaces with specific curvature speed.
result The flow becomes strictly h-convex and converges to a geodesic sphere.
Researchers find H-convex functions for Heisenberg group sets.
problem Finding H-convex functions for H-convex sets in the Heisenberg group.
method Extension of Fenchel's convex family concept, employing precise conditions.
result Conditions on set shape for existence of H-convex functions.
Having in mind the well known model of Euclidean convex hypersurfaces [4], [5], and the ideas in [1] many authors defined and investigate convex hypersurfaces of a Riemannian manifold. As it was proved by the first author in [7], there follows the interdependence between convexity and Gauss curvature of the hypersurfac…
The paper studies curvature flows in hyperbolic space and proves geometric inequalities.
problem Proving geometric inequalities in hyperbolic space using curvature flows.
method Locally constrained curvature flows, h-convexity, and shifted principal curvatures.
result Established new sharp geometric inequalities comparing curvature integrals to quermassintegrals.
Given a real-valued function c c c defined on the cartesian product of a generic Carnot group $\G$ and the first layer V 1 V_1 V 1 of its Lie algebra, we introduce a notion of c c c horizontal convex ( c c c H-convex) function on $\G$ as the supremum of a suitable family of affine functions; this family is defined pointwisely, and …
Maximal distortion between geodesic and Euclidean diameters in polygonal domains is studied.
problem Maximal ratio of geodesic to Euclidean diameters in polygonal domains with holes.
method Analyzes convex polygons with holes, using geometric triangulations as a comparison.
result The supremum of the ratio is between Ω ( h 1 / 3 ) Ω(h^{1/3}) Ω ( h 1/3 ) and O ( h 1 / 2 ) O(h^{1/2}) O ( h 1/2 ) for convex polygons. Paper solves Christoffel-Minkowski problem in hyperbolic space.
problem Prescribing k k k -th horospherical p p p -surface area measure of h h h -convex domains in hyperbolic space. method Considered a fully nonlinear equation and used the full rank theorem with a viscosity approach.
result Existence of uniformly h h h -convex solution under appropriate assumptions. New inequalities derived for hyperbolic space via specific flows.
problem Sharp inequalities for mean and k-th mean curvatures in hyperbolic space.
method Locally constrained inverse curvature flow by Brendle, Guan, and Li.
result Established and verified new sharp inequalities for hyperbolic space.
Proves new inequality for hyperbolic space hypersurfaces.
problem Finding inequalities for hypersurfaces in hyperbolic space.
method Proves a Heintze-Karcher type inequality for shifted mean convex hypersurfaces.
result Proves Alexandrov type theorem and uniqueness result for hypersurfaces.
New method for optimization on Hadamard manifolds with curvature-independent guarantees.
problem Curvature-dependent complexity in geodesic convex optimization.
method Introducing horospherical convexity and developing algorithms for optimization.
result Curvature-independent convergence of subgradient descent and Nesterov's method.
Study extends convexity in curved spaces using fractional integrals.
problem Extending convexity to curved spaces with nonpositive curvature.
method Introducing (geodesically) h h h -convex functions and using Katugampola's fractional integrals. result Essentially sharp estimate involving squared distance mappings.
This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.
problem Prescribing curvatures for convex hypersurfaces in hyperbolic space.
method Proving a full rank theorem to establish the existence of solutions.
result Existence of solutions to the Christoffel problem and its equivalent Nirenberg-Kazdan-Warner problem on spheres.
The paper proves Michael-Simon inequalities in hyperbolic space using novel curvature flows.
problem Proving the sharp Michael-Simon inequality for mean curvature in hyperbolic space.
method Developed new locally constrained curvature flows for proving the inequality.
result Sharp Michael-Simon inequalities for mean and k-th mean curvatures in starshaped hypersurfaces in hyperbolic space.