arXiv research

The paper studies quasi-$X$-convex functions and their applications in optimization.

Unified framework for robust risk measures beyond convexity.

Study cash-subadditive risk measures without quasi-convexity.

Paper introduces quasi-logconvex risk measures and their properties.

The paper defines quasi-convex subsets in spaces with lower curvature bound.

Study shows infimum of dual volume equals convex core volume for hyperbolic 3-manifolds.

We show that for every simple closed curve α, the extremal length and the hyperbolic length of αare quasi-convex functions along any Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space equipped with the Teichmuller metric, balls are quasi- convex.

Sharp estimates for Finsler metrics in convex domains.

It is known that every infinite index quasi-convex subgroup $H$ of a non-elementary hyperbolic group $G$ is a free factor in a larger quasi-convex subgroup of $G$. We give a probabilistic generalization of this result. That is, we show that when $R$ is a subgroup generated by independent random walks in $G$, then $\lan…

The paper studies $m$-quasi Einstein manifolds with convex potential and finds constant scalar curvature.

In this article, we study convex affine domains which can cover a compact affine manifold. For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least $C^1$ boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional par…

Paper infers intrinsic dimension from quasi-convex measurements.

We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of $Out(F_n)$ as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via …

Maps between Hadamard manifolds are quasi-isometric to harmonic maps.

Ricci curvature links volume convexity and minimal submanifolds.

Geodesic flows on specific manifolds are structurally stable.

This expository paper presents elementary proofs of four basic results concerning derivatives of quasi-convex functions. They are combined into a fifth theorem which is simple to apply and adequate in many cases. Along the way we establish the equivalence of the basic lemmas of Jensen and Slodkowski.

New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.

Suppose $τ$ is a train track on a surface $S$. Let $C(τ)$ be the set of isotopy classes of simple closed curves carried by $τ$. Masur and Minsky [2004] prove $C(τ)$ is quasi-convex inside the curve complex $C(S)$. We prove the complement, $C(S) - C(τ)$, is quasi-convex.

In recent work, the notion of Double Convexity for a foliation of a conical null hypersurface was introduced to give a proof, if satisfied, of the Null Penrose Inequality. Double Convexity constrains the geometry of a Marginally Outer Trapped Surface (MOTS), called a quasi-round MOTS. In the first part of this paper, f…

New concept of coarse medians for higher rank symmetric spaces.

Develops theory of relatively geometric actions on CAT(0) cube complexes.

This paper extends boundary embedding results to coarsely convex spaces.

Study convex embeddability in linear and circular orders, applying to knots.

The study characterizes subgroups of mapping tori of free groups.

Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.

Study on Cheeger constant in specific hyperbolic 3-manifolds.

Study of groups and their quasi-isometrically embedded subgroups.

Study of hyperbolic directions in convex projective geometry.

Making use of the dual Bonahon-Schläfli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold $M$ is bounded by an explicit constant, depending only on the topology of $M$, times the Weil-Petersson distance between the hyperbolic structures on the upper and lower boundary components of …

We prove that, for every closed (not necessarily convex) hypersurface $Σ$ in $\mathbb{R}^{n+1}$ and every $p>n$, the $L^p$-norm of the trace-free part of the anisotropic second fundamental form controls from above the $W^{2,p}$-closeness of $Σ$ to the Wulff shape. In the isotropic setting, we provide a simpler proof. T…

Recent studies incorporate Nesterov's accelerated gradient method for the acceleration of gradient based training. The Nesterov's Accelerated Quasi-Newton (NAQ) method has shown to drastically improve the convergence speed compared to the conventional quasi-Newton method. This paper implements NAQ for non-convex optimi…

New method solves convex optimization faster than NAG.

We show that any infinite order element $g$ of a virtually cyclic hyperbolically embedded subgroup of a group $G$ is Morse, that is to say any quasi-geodesic connecting points in the cyclic group $C$ generated by $g$ stays close to $C$. This answers a question of Dahmani-Guirardel-Osin. What is more, we show that hyper…

The main point of this paper is to prove the following useful result: If the almost everywhere 2-jet of a locally quasi-convex function u satisfies a degenerate elliptic constraint F, then u is F-subharmonic, i.e., u is a viscosity F-subsolution. This AE Theorem makes otherwise difficult results transparent. Some insta…

The paper constructs convex subsets in anti-de Sitter space with specific metrics on boundaries.

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes span…

In [19], a general, inexact, efficient proximal quasi-Newton algorithm for composite optimization problems has been proposed and a sublinear global convergence rate has been established. In this paper, we analyze the convergence properties of this method, both in the exact and inexact setting, in the case when the obje…

We show that for $X$ a proper $\mathrm{CAT}(-1)$ space there is a maximal open subset of the horofunction compactification of $X\times X$ with respect to the maximum metric that compactifies the diagonal action of an infinite quasi-convex group of the isometries of $X$. We also consider the product action of two quasi-…

The paper proves convexity results for a specific type of Lie groups.

Study growth rates of subgroups in groups with a constricting element.

Nous considérons un espace topologique qui est localement isomorphe au quotient de R^k par l'action d'un groupe discret et nous l'appelons quasi-variété de dimension k. Les quasi-variétés généralisent les variétés et les V-variétés et représentent le cadre naturel pour la réduction symplectique par rapport à l'action i…

New quasi-Newton method guarantees global superlinear convergence.

We define an explicit quasi-local mass functional which is non-decreasing along all foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the null Penrose conjecture under fairly generic conditions.

New insights into risk aversion for complex decision models.

The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.

Novel algorithm accelerates PnP methods for image deblurring and super-resolution.

We interpret the variational inference of the Stochastic Gradient Descent (SGD) as minimizing a new potential function named the \textit{quasi-potential}. We analytically construct the quasi-potential function in the case when the loss function is convex and admits only one global minimum point. We show in this case th…