arXiv research

The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.

The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.

Generalizes Fenchel conjugation to nonlinear functions on arbitrary sets.

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…

The paper generalizes Fenchel's theorem for curves with singularities.

Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.

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…

We generalize the Fenchel theorem to strong spacelike (which means that the tangent vector and the curvature vector span a spacelike 2-plane at each point) closed curves with index 1 in the 3-dimensional Lorentz space, showing that the total curvatures must be less than or equal to $2π$. A similar generalization of the…

The article proves inequalities for capillary hypersurfaces in hyperbolic space.

New dual formulation reduces generalization error for ERM-fDR.

Defines Fenchel-Nielsen coordinates for SL(3,C) representations.

At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes…

Dual optimization connects ERM-fDR to normalization function.

We study the convex duality method for robust utility maximization in the presence of a random endowment. When the underlying price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true for a wide class of utility functions on the whole real line and unbounded random endo…

Calculates twist in Teichmüller space using cross ratios.

Paper offers a dual formulation for consumption problem with multiplicative habit.

Introduces Fitzpatrick losses, tighter than Fenchel-Young losses.

Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.

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.

Study flows to analyze sphere quermassintegrals.

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.

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 proves a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.

In this paper we investigate the relation between complexified Fenchel-Nielsen coordinates and spectral network coordinates on Seiberg-Witten moduli space. The main technique is the comparison of exact expressions for the expectation value of 't Hooft defects in certain 4D $SU(2)$ $\mathcal{N}=2$ gauge theories. We der…

The paper computes Fenchel-Nielsen coordinates for fixed points of cyclic actions on Teichmüller space.

The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $m^{\H}_k$, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we pr…

Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition $\mathcal{P}$ 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.

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.

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, $n \geq 3$.

Paper solves inequalities for capillary hypersurfaces in half-spaces.

We provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel-Nielsen …

Two flexible, degenerate constructions related to Thurston's theorem.

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.

Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.

Study geodesic flow on symmetric surfaces to determine parabolic type.

Paper extends Fenchel's conjecture to non-Euclidean crystallographic groups.

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…

New comparison theorem for submanifolds with geometric inequalities.

Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.

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.

The paper proves stability of inequalities for nearly spherical sets in various spaces.