The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.
problem Defining Fenchel conjugate and biconjugate on curved spaces.
method Introduced a new definition of Fenchel conjugate and biconjugate on Hadamard manifolds based on the tangent bundle.
result Developed a Fenchel-Moreau Theorem for geodesically convex functions on Hadamard manifolds.
Defines Fenchel-Nielsen coordinates for SL(3,C) representations.
problem No specific problem stated; coordinates defined for a new context.
method Introduced Fenchel-Nielsen coordinates for mSL(3,C) representations. result Relates to classical and generalized Fenchel-Nielsen coordinates.
Generalizes Fenchel conjugation to nonlinear functions on arbitrary sets.
problem Extending Fenchel conjugation to functions on arbitrary sets without structure.
method Replacing linear test functions with nonlinear ones, investigating properties including biconjugation.
result Derived further results on smooth manifolds and Lie groups, relating to convexity.
The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.
problem Finding minimizers of nonlocal curvature energies.
method Combining Fenchel-type theorems with geometric analysis techniques.
result Circles and disks minimize specific energy functionals.
Calculates twist in Teichmüller space using cross ratios.
problem Calculating the Fenchel-Nielsen twist in Teichmüller space.
method Using cross ratio coordinates.
result Compact calculation of twist in Teichmüller space.
Introduces Fitzpatrick losses, tighter than Fenchel-Young losses.
problem Improving loss functions for machine learning.
method Introduces Fitzpatrick losses based on the Fitzpatrick function.
result Fitzpatrick losses are tighter than Fenchel-Young losses.
Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.
problem Developing inequalities for submanifolds in curved spaces.
method Connecting Fenchel-Willmore and logarithmic Sobolev inequalities for mean-convex submanifolds.
result Established extensions of Fenchel-Willmore inequality and derived new Sobolev-type inequalities.
Various Alexandrov-Fenchel type inequalities have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. The purpose of this article is to shed new light on this by c…
Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
problem Relationship between Fenchel-Nielsen coordinates and shear coordinates on Riemann surfaces.
method Explicitly showed the Poisson bracket on shear coordinates induces the Fenchel-Nielsen bracket on gluing parameters.
result Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
Study flows to analyze sphere quermassintegrals.
problem Analyze quermassintegrals on the sphere.
method Use two types of flows to study quermassintegrals.
result Establish Alexandrov-Fenchel inequalities for the sphere.
This paper studies Fenchel-Young losses, a generic way to construct convex loss functions from a regularization function. We analyze their properties in depth, showing that they unify many well-known loss functions and allow to create useful new ones easily. Fenchel-Young losses constructed from a generalized entropy, …
Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.
problem Establishing a Fenchel-Willmore inequality for submanifolds in manifolds with non-negative Ricci curvature.
method Analyzing submanifolds in manifolds with non-negative intermediate Ricci curvature and Euclidean volume growth.
result Sharp Fenchel-Willmore inequality for submanifolds in manifolds with non-negative intermediate Ricci curvature.
Given a real-valued function defined on the Heisenberg group, we provide a definition of abstract convexity and Fenchel transform that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the Euclidean case, a convex function can be characterized via its iterated Fen…
Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.
problem Extending capillary convex body results to anisotropic setting.
method Developed theory for anisotropic capillary convex bodies in half-space and established Alexandrov-Fenchel inequality for mixed volumes.
result Established a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies, weakening and extending previous results.
Paper proves a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
problem Proving a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
method Using a locally constrained nonlinear curvature flow to preserve the n-th quermassintegral and decrease the k-th quermassintegral. result Obtained the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in Bn+1. The paper computes Fenchel-Nielsen coordinates for fixed points of cyclic actions on Teichmüller space.
problem Computing Fenchel-Nielsen coordinates for cyclic actions on Teichmüller space.
method Developed algorithms to describe Fenchel-Nielsen coordinates of fixed points of cyclic subgroups of Mod(S_g) on Teich(S_g).
result Computed Fenchel-Nielsen coordinates for cyclic subgroups of orders 10, 8, and 4 in Mod(S_2).
Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition P and given a base complex structure X on S, there is an associated deformation space of complex structures on S, which we call the Fenchel-Nielsen Teichmüller space associated to the pair $(\…
Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.
problem Proving Wolpert's formula for the Weil-Petersson symplectic form.
method Introducing a cell decomposition and groupoid cocycle on a surface to represent points in Teichmüller space.
result Topological proof of Wolpert's formula for the Weil-Petersson symplectic form.
We extend the classical Aleksandrov-Fenchel inequality for mixed volumes to functionals arising naturally in hermitian integral geometry. As a consequence, we obtain Brunn-Minkowski and isoperimetric inequalities for hermitian quermassintegrals.
Study flow on de Sitter space for convex hypersurfaces.
problem Behavior of locally constrained inverse curvature flow in de Sitter space.
method Analyze flow on de Sitter space with specific initial conditions and inequalities.
result Derive Alexandrov-Fenchel type inequalities.
We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov-Fenchel-type inequality for strictly convex hypersurfaces in the n-dimensional sphere, n≥3.
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.
In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space $\H^n$, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov-Fenchel inequalities.
We develop Fenchel-Nielsen coordinates for representations of surface groups into Sp(2n,R) with maximal Toledo invariant. Analogous to classical Fenchel-Nielsen coordinates on the Teichmüller space they consist of a parametrization of representations of the fundamental group of a pair of pants and a careful investigati…
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.
The article proves inequalities for capillary hypersurfaces in hyperbolic space.
problem Proving inequalities for capillary hypersurfaces in hyperbolic space.
method Constructing a new locally constrained inverse curvature flow.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space.
Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.
problem Understanding the evolution of hypersurfaces with capillary boundaries.
method Locally constrained mean curvature flow for star-shaped hypersurfaces in the half-space.
result Established new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary.
Study geodesic flow on symmetric surfaces to determine parabolic type.
problem Determine conditions for a Riemann surface to be of parabolic type.
method Analyze Fenchel-Nielsen coordinates and covering group properties.
result Conditions for a surface to be parabolic are equivalent to specific properties of its Fenchel-Nielsen coordinates.
Paper extends Fenchel's conjecture to non-Euclidean crystallographic groups.
problem Does every non-Euclidean crystallographic group have a specific subgroup?
method Examined various cases of non-Euclidean crystallographic groups and their orbit spaces.
result Affirmative answer in many cases, open in others.
Over the past decades, numerous loss functions have been been proposed for a variety of supervised learning tasks, including regression, classification, ranking, and more generally structured prediction. Understanding the core principles and theoretical properties underpinning these losses is key to choose the right lo…
Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.
problem Deriving upper bounds for the Alexandrov-Fenchel deficit.
method Using weighted Minkowski integral formulas and an integral formula for the deficit in Jensen's inequality.
result Quantitative estimates under weaker convexity assumptions, including a distance term.
We cut a hyperbolic surface of finite area along some analytic simple closed curves, and glue in cylinders of varying moduli. We prove that as the moduli of the glued cylinders go to infinity, the Fenchel-Nielsen twist coordinates for the resulting surface around those cylinders converge.
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 paper proves stability of inequalities for nearly spherical sets in various spaces.
problem Stability of geometric inequalities for nearly spherical sets.
method Deriving a quantitative quermassintegral inequality and applying it to derive stability results.
result Stability of geometric inequalities involving weighted curvature integrals and quermassintegrals for nearly spherical sets in Rn+1 and Hn+1. Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.
problem Geometric inequalities and mixed Hodge-Riemann relations for translation-invariant valuations.
method Proves mixed Hodge-Riemann relations for various convex bodies and their mixed volumes.
result Strengthened geometric inequalities for lower dimensional convex bodies.
Researchers define new quantum representations for a Lorentz algebra and study their Clebsch-Gordan decomposition.
problem Quantum representations of a Lorentz algebra and their Clebsch-Gordan decomposition.
method Defined new infinite-dimensional irreducible representations using quantum torus algebra and quantized Chern-Simons theory.
result The Clebsch-Gordan decomposition of tensor product representations reduces to problems in Fenchel-Nielson length operators in quantized Chern-Simons theory.
Kahn and Markovic \cite{KahnMark} proved that the fundamental group of each closed hyperbolic three manifold contains a closed surface subgroup. One of the main ingredients in their proof is a theorem which states that an assignment of nearly real, complex Fenchel-Nielsen coordinates to the cuffs of a pants decompositi…
We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abe…
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.
We generalize the Fenchel theorem for strong spacelike closed curves of index 1 in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to 2π. Here strong spacelike means that the tangent vector and the curvature vector span a spacelike 2-plane at each point of the curve $γ…
In this paper we combine our recent work on regular globally hyperbolic maximal anti-de Sitter structures with the classical theory of globally hyperbolic maximal Cauchy-compact anti-de Sitter manifolds in order to define an augmented moduli space. Moreover, we introduce a coordinate system in this space that resembles…
In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…
The paper improves inequalities for nearly spherical sets using quermassintegrals.
problem Improving inequalities for nearly spherical sets.
method Establishing quantitative Alexandrov-Fenchel inequalities for quermassintegrals.
result Lower bounds on the (k,m)-isoperimetric deficit found using spherical deviation and asymmetry. Flow of convex hypersurfaces in hyperbolic space converges to geodesic spheres.
problem Understanding the evolution of convex hypersurfaces in hyperbolic space.
method Gauss curvature type flow, Alexandrov-Fenchel inequality application.
result Smooth solution converges to geodesic spheres.
New method uses FY loss for better inverse optimization.
problem Estimating unknown parameters from noisy and suboptimal solutions.
method Fenchel-Young loss approach for efficient gradient-based optimization.
result Significant improvement in parameter estimation accuracy and computational speed.
We consider complex Fenchel-Nielsen coordinates on the quasi-Fuchsian space of punctured tori. These coordinates arise from a generalisation of Kra's plumbing construction and are related to earthquakes on Teichmueller space. They also allow us to interpolate between two coordinate systems on Teichmueller space, namely…
The paper generalizes Fenchel's theorem for curves with singularities.
problem Proving a generalized Fenchel's theorem for closed curves with singularities.
method Generalization of Fenchel's theorem for closed frontal curves in Euclidean space.
result Total absolute curvature of non-co-orientable closed frontal curves is at least π, with equality conditions.
We discuss bases of the space of holomorphic quadratic differentials that are dual to the differentials of Fenchel-Nielsen coordinates and hence appear naturally when considering functions on the set of hyperbolic metrics which are invariant under pull-back by diffeomorphisms, such as eigenvalues of the Laplacian. The …