For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…
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48 results for “spherical convexity”
Radial expansion preserves hyperbolic convexity and radial contraction preserves spherical convexitymath.DG
Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.
problem Preserving convexity in hyperbolic and spherical geometries under radial transformations.
method Used Poincaré disk model for hyperbolic geometry and stereographic projection for spherical geometry to prove preservation of convexity under radial expansion and contraction.
result Radial expansion and contraction preserve hyperbolic and spherical convexity, respectively.
Study spherical convex bodies using -floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced -floating areas and curvature entropy for spherical convex bodies.
result Established isoperimetric inequalities and dual isoperimetric inequalities.
Paper investigates curvature problems and existence of solutions.
problem Existence of admissible solutions to curvature problems.
method Investigates curvature problems with prescribed quotient type, proving existence under specific conditions.
result Proves existence of admissible solutions without additional conditions.
A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a connected (n-1)-manifold into the n-sphere is a surjection onto the boundary of a c…
New capillary Christoffel-Minkowski problem solved for half-space.
problem Finding strictly convex capillary hypersurfaces from capillary functions.
method Established analogous result in capillary setting for half-space.
result Capillary functions lead to strictly convex capillary hypersurfaces.
The paper studies a flow for convex capillary hypersurfaces in a ball, proving smooth convergence to a spherical cap.
problem Analyzing the behavior of convex capillary hypersurfaces under mean curvature flow.
method Introduced mean curvature flow for hypersurfaces in the unit Euclidean ball with capillary boundary. Proved smooth convergence to a spherical cap for strictly convex initial hypersurfaces.
result The flow preserves strict convexity and converges smoothly to a spherical cap for all positive time.
Paper extends Schur's theorem to spherical curves via monotonicity.
problem Comparing chord lengths of convex and spherical curves.
method Monotonicity and expansion module approach.
result Schur's Theorem extended to spherical curves.
Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.
problem Prescribed curvature problem for convex capillary hypersurfaces.
method Reformulated as Hessian quotient equation with Robin boundary condition.
result Existence of strictly convex capillary hypersurface with prescribed curvature.
The paper extends geometric inequalities for nearly spherical sets in various space forms.
problem Investigating weighted inequalities for nearly spherical sets in space forms.
method Generalizing and extending inequalities for nearly spherical sets in and settings, with convex weight functions.
result Quantitative stability estimates for weighted inequalities in and .
Optimal inequality on sphere for convex bodies.
problem Bounding the spherical measure of a convex body's intersection with a plane orthogonal to its centroid.
method Proving an inequality on the sphere using convex geometry.
result The inequality is optimal with a constant of \((1-1/n)^{n-1}\).
Let B be a thick spherical building equipped with its natural CAT(1) metric and let M be a proper, convex subset of B. If M is open or if M is a closed ball of radius pi/2, then the maximal subcomplex supported by the complement of M is spherical and non contractible.
New inequalities for convex hypersurfaces in various spaces.
problem Deriving inequalities for hypersurfaces under convex weight.
method Sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces in Euclidean, spherical, and hyperbolic spaces.
result Incorporates convex, non-decreasing positive functions as weights, yielding a broad family of geometric inequalities.
A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…
Compact proof for Brown-York mass positivity and rigidity in flat and spherical spaces.
problem Positivity of Brown-York mass and rigidity of manifolds with mean-convex boundaries.
method Spinorial proof and optimal lower bound for eigenvalues.
result Optimal lower bound for first non-null eigenvalue of Dirac operator.
The paper explores centroids and static equilibrium points in non-Euclidean geometries.
problem Investigating centroids and static equilibrium points in spherical, hyperbolic, and normed spaces.
method Extending Gal'perin's work, the paper examines convex bodies in these spaces and analyzes the minimum number of equilibrium points.
result Every plane convex body in any of these spaces has at least four equilibrium points, and there are mono-monostatic convex bodies in 3D spherical, hyperbolic, and certain normed spaces.
Conformal qc geometry of spherical qc manifolds are investigated. We construct the qc Yamabe operators on qc manifolds, which are covariant under the conformal qc transformations. A qc manifold is scalar positive, negative or vanishing if and only if its qc Yamabe invariant is positive, negative or zero, respectively. …
Discrete Laplacians defined for spherical and hyperbolic surfaces.
problem Defining discrete Laplacians for non-Euclidean geometries.
method Definitions close to Euclidean, structure-preserving properties proven.
result Connection between discrete and smooth Laplacians in non-Euclidean settings.
We show non-collapsing for the evolution of nearly spherical closed convex curves in \mathbb{R}^2 under power curvature flow using two-point-methods.
We consider a compact, star-shaped, mean convex hypersurface . We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which …
New methods optimize functions on hyperbolic and spherical spaces, matching Euclidean rates up to logarithmic factors.
problem Optimizing functions on non-Euclidean spaces like hyperbolic and spherical geometries.
method Introduced accelerated global first-order methods for -smooth and geodesically convex functions on hyperbolic and spherical spaces.
result Achieved the same rates as accelerated gradient descent in Euclidean space, up to logarithmic factors.
The paper studies K-stability of spherical varieties and their degenerations.
problem Understanding K-stability and degenerations of polarized spherical varieties.
method Reduction to a variational problem on the moment polytope, convexity constraint, and solving the HMA equation.
result Determines strict semistability and polystable degenerations for Fano spherical varieties of rank two.
Proof of Graustein's theorem in different geometries.
problem Average curvature of plane ovals and convex curves in various geometries.
method Wave propagation approach for different geometries.
result The average curvature is attained at least at four points in different geometries.
We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This…
Unique metric found for discrete curvature on spherical cone-metrics.
problem Finding a unique metric with prescribed curvature on spherical cone-metrics.
method Discrete conformal approach to spherical cone-metrics.
result Existence of a unique metric realizing prescribed curvature in each conformal class.
The paper extends Liebmann's Theorem to convex hypersurfaces with boundary.
problem Proving properties of convex hypersurfaces with boundary in Euclidean space.
method Analyzing locally convex, embedded, compact, connected CMC hypersurfaces bounded by a closed strictly convex submanifold.
result Spherical caps are the only such hypersurfaces with non-zero constant mean curvature bounded by a sphere.
Study on spherical bodies of constant width on the unit sphere, proving bounds on their relative effective radius.
problem Understanding the smallest spherical bodies of constant width on the unit sphere.
method Analyzing spherical bodies of constant width on the unit sphere, constructing examples and applying geometric arguments.
result Proved non-trivial bounds on the relative effective radius of spherical bodies of constant width.
Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundarymath.DG
Paper solves inequalities for capillary hypersurfaces in half-spaces.
problem Finding inequalities for convex capillary hypersurfaces in half-spaces.
method Introduced quermassintegrals and constructed a new locally constrained curvature flow to prove convergence to spherical caps.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
Classifies 3-manifolds with uniformly positive scalar curvature.
problem Classifying 3-manifolds with uniformly positive scalar curvature.
method Analyzes properties of 3-manifolds with mean convex boundaries and uniformly positive scalar curvature.
result 3-manifolds with uniformly positive scalar curvature are homeomorphic to sums of spherical 3-manifolds and .
The paper classifies and studies symplectic and contact properties of circular spherical divisors.
problem Investigating symplectic and contact topology of circular spherical divisors.
method Classification and analysis of concave circular spherical divisors, including embedding, Stein fillability, and rational homology type determination.
result All concave circular spherical divisors up to toric equivalence are realized as symplectic log Calabi-Yau pairs with minimal complements.
New findings on hyperbolic groups and their boundaries.
problem Understanding the structure of cubulated hyperbolic groups with specific boundary conditions.
method Utilizing ideas from Markovic's work on Cannon's conjecture, focusing on quasi-convex subgroups and limit sets.
result Cubulated hyperbolic groups with certain boundary conditions are virtually fundamental groups of specific manifolds.
The paper proves inequalities for convex capillary hypersurfaces in a half-space.
problem Proving inequalities for convex capillary hypersurfaces in a half-space.
method Locally constrained inverse curvature flow with spherical cap convergence.
result Proves a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
The paper simplifies K-stability conditions for spherical varieties.
problem K-stability of polarized spherical varieties.
method Expressed K-stability in combinatorial terms, provided sufficient conditions.
result G-uniform K-stability provides a checkable condition for existence of constant scalar curvature metrics.
New interpretation of discrete conformality using polyhedral convex hulls.
problem Understanding discrete conformality in 3D.
method Epstein-Penner convex hull construction and induced metrics.
result New bijections and interpretations of discrete conformality.
It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed…
Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flowmath.DG
Study shows how curved surfaces evolve smoothly to spherical shapes.
problem Evolution of curved surfaces with capillary boundaries.
method Volume-preserving curvature flow with power mean curvature speed.
result Convex initial hypersurfaces evolve to spherical caps over time.
Paper solves capillary Orlicz-Minkowski problem with new inequalities.
problem Finding capillary convex bodies with prescribed Orlicz surface area measures.
method Continuity method and inequalities to solve the capillary even Orlicz-Minkowski problem.
result Volume-normalized smooth solutions and inequalities established.
Ancient convex solutions to flow equations are limited to simple shapes.
problem Characterizing ancient convex solutions to flow equations.
method Analyzing mean curvature flow and curvature functions of convex hypersurfaces.
result Ancient convex solutions to flow equations are limited to spherical, cylindrical, or planar shapes.
Fixed points of mean section operators found in convex bodies.
problem Characterizing fixed points of mean section operators in convex bodies.
method Characterization of rotation equivariant operators using spherical Laplacian mass distribution, and application of Minkowski valuations.
result Euclidean balls are the only fixed points of mean section operators in a neighborhood of the unit ball.
We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant on the space of those isometric deformations which, for conv…
It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if…
Stability of Convex Disksmath.DG
Proves stability of convex disks close to round caps.
problem Stability of convex disks with positive curvature and strictly convex boundary.
method Compactness result for a Liouville-type PDE problem.
result Proves stability for a theorem of F. Hang and X. Wang.
The paper proves new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
problem Proving new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
method Locally constrained inverse curvature flows in hyperbolic and spherical spaces.
result Established new Alexandrov-Fenchel and Minkowski inequalities involving general convex weight functions.
The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean an…
Alexandrov-Fenchel type inequalities for convex hypersurfaces in hyperbolic space and in spheremath.DG
In this paper, firstly, inspired by Natário's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the sphere $\SS^{n+1}$. We also get the rigidity in the spherical case. Secondly, we use …
Study on convex capillary hypersurfaces with Lp curvature in half-space.
problem Prescribed Lp curvature for convex capillary hypersurfaces.
method Reduction to Hessian quotient equation with Robin boundary condition.
result Existence and uniqueness of smooth admissible solutions.
Study concavity of solutions to elliptic equations under conformal deformations.
problem Establish concavity estimates for the principle eigenfunction of weighted Schrödinger operators.
method Analyzing the Dirichlet problem for the weighted Schrödinger operator \[-Δu + Vu = λρu\] with conformal connections.
result Partial resolution of Nguyen's conjecture on fundamental gap of horoconvex domains and power convexity estimate for solutions in spherical geometry.
The spherical Radon transform on the unit sphere can be regarded as a member of the analytic family of suitably normalized generalized cosine transforms. We derive new formulas for these transforms and apply them to study classes of intersections bodies in convex geometry.