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.
Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.
problem Anisotropic non-homogeneous Gauss curvature flows and Orlicz-Minkowski problems.
method Long-time existence and behavior analysis, parabolic approximation method, curvature flow.
result Existence and new results for Orlicz-Minkowski problems, including Lp versions. New method solves generalized Minkowski problem for torsional rigidity.
problem Generalized Minkowski problem for torsional rigidity.
method Flow method
result Existence of solutions for general measures.
Study of a flow related to the Orlicz-Minkowski problem for convex hypersurfaces.
problem Orlicz-Minkowski problem involving Gauss curvature and support function.
method Generalized Gauss curvature flow for convex hypersurfaces in Euclidean n-space.
result Long-time existence and convergence of the flow, leading to existence results for the Orlicz-Minkowski problem.
New method solves a generalized Minkowski problem using a curvature flow.
problem Generalized Minkowski problem for smooth measures.
method Flow involving Gauss curvature and support function.
result Existence of solutions for the dual Orlicz-Minkowski problem.
Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.
problem Solving dual Orlicz Minkowski problems for anisotropic flows.
method Anisotropic inverse Gauss curvature flows and stationary solutions.
result New existence results for dual Orlicz Minkowski problems for smooth measures.
In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the …
Study an anisotropic capillary flow to solve capillary Orlicz-Minkowski problem.
problem Solve capillary Orlicz-Minkowski problem without evenness assumption.
method Analyze an anisotropic capillary Gauss curvature flow to prove convergence and establish existence.
result Establish existence result for capillary Orlicz-Minkowski problem without evenness assumption.
Study anisotropic flows without global terms and solve dual Orlicz Christoffel-Minkowski problems.
problem Anisotropic flows without global forcing terms and dual Orlicz Christoffel-Minkowski problems.
method Existence results for dual Orlicz Christoffel-Minkowski type problems via stationary solutions of anisotropic flows.
result Existence results for a class of dual Orlicz Christoffel-Minkowski type problems.
The general volume of a star body, a notion that includes the usual volume, the qth dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the (p,q)-dual curvature measures introduced recently by Lutwak, Ya…
Smooth even solutions found for a generalized convex geometry problem.
problem Dual Orlicz-Minkowski problem in convex geometry.
method Geometric flow involving Gauss curvature and normal vectors.
result Existence of smooth even solutions for smooth even measures.
This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz φ-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz φ-radial addition of two star bodies, we derive a f…
This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz-Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is…
Paper solves Minkowski problem for q-torsional rigidity using curvature flow.
problem Solving Minkowski problem for q-torsional rigidity.
method Method of curvature flows.
result Existence of smooth non-even solutions.
Paper solves a generalized chord Minkowski problem using Gauss curvature flows.
problem Generalized chord Minkowski problem using Orlicz functions.
method Gauss curvature flows to achieve existence of smooth solutions.
result Existence of smooth solutions to the orlicz chord Minkowski problem.
Paper solves Musielak-Orlicz-Gauss image problem using parabolic flows.
problem Characterize Musielak-Orlicz-Gauss image measure of convex bodies.
method Study of parabolic flows and approximation technique.
result Provides solutions to the extended Musielak-Orlicz-Gauss image problem.
Study proves only origin-centered spheres solve certain curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Using the Heintze-Karcher inequality, the study proves the uniqueness of smooth, strictly convex solutions to a class of Minkowski type problems.
result Only origin-centered spheres solve isotropic and Lp-Gaussian-Minkowski problems. This paper aims to develop basic theory for the dual Orlicz Lφ affine and geominimal surface areas for star bodies, which belong to the recent dual Orlicz-Brunn-Minkowski theory for star bodies. Basic properties for these new affine invariants will be provided. Moreover, related Orlicz affine isoperimetric inequalit…
The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the Lp-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz Lφ affine and geominimal surface areas for single convex body as well as for multiple convex bod…
The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies K in Rn containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more gener…
Paper proves rigidity results for anisotropic capillary hypersurfaces.
problem Rigidity of anisotropic capillary hypersurfaces.
method New Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces.
result Uniqueness of solution to anisotropic Orlicz-Christoffel-Minkowski problem.
New Orlicz Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are…
This paper solves the dual Minkowski problem for q-torsional rigidity.
problem The dual Minkowski problem for q-torsional rigidity.
method Introduced the p-th dual q-torsional measure and solved the p-th dual Minkowski problem for q-torsional rigidity using a Gauss curvature flow.
result Existence of smooth even and non-even solutions to the p-th dual Minkowski problem for q-torsional rigidity.
Paper solves Orlicz-Aleksandrov problem using Gauss curvature flow.
problem Orlicz-Aleksandrov problem
method Gauss curvature flow
result Existence of smooth solutions
The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.
problem Deforming convex hypersurfaces in Euclidean space.
method Fully nonlinear curvature flow involving k-th elementary symmetric function and support function.
result Long-time existence and convergence of the flow under certain assumptions.
A new method for transporting unbalanced measures on graphs efficiently.
problem Optimal transport for measures with unequal total masses on graph metric spaces.
method Developed a novel variant of entropy partial transport (Orlicz-EPT) with Orlicz geometric structure, leading to Orlicz-Sobolev transport (OST).
result OST can be efficiently computed by solving a univariate optimization problem, significantly faster than Orlicz-EPT.
The paper proves uniqueness of solutions to curvature problems using various methods.
problem Proving uniqueness of solutions to anisotropic and isotropic curvature problems.
method Integral formulas by S. S. Chern and Simon's uniqueness result, along with new methods.
result The only smooth strictly convex solution to the isotropic curvature problem is an origin-centred sphere.
We carry out calculations of Orlicz cohomology for some basic Riemannian manifolds (the real line, the hyperbolic plane, the ball). Relationship between Orlicz cohomology and Poincaré--Sobolev--Orlicz-type inequalities is discussed.
A new statistical model uses Orlicz-Sobolev spaces with Gaussian weight.
problem Statistical modeling of infinite-dimensional probability measures.
method Affine statistical bundle on Gaussian Orlicz-Sobolev space.
result Provides tools for solving infinite-dimensional evolution problems.
Study timelike Ricci curvature bounds via optimal transport with Orlicz-type costs.
problem Characterize timelike Ricci curvature bounds.
method Optimal transport with Orlicz-type costs, convexity of relative entropy.
result Characterize timelike Ricci curvature lower bounds via convexity of relative entropy.
This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, …
SL(n) covariant valuations on Orlicz spaces are represented and characterized.
problem Representing SL(n) covariant valuations on Orlicz spaces.
method Representation theorem established for continuous, SL(n) covariant vector-valued valuations.
result Unique characterization of SL(n) covariant valuations as moment vectors.
Introduces generalized Orlicz premia for broader applicability.
problem Developing a flexible framework for insurance premium calculation.
method Introduces a generalized Orlicz premium definition using non-convex loss functions.
result Generalized Orlicz premia encompass various specific cases and maintain key properties.
Classification of SL(n) covariant valuations on Orlicz spaces.
problem Classifying continuous SL(n) covariant valuations on Orlicz spaces.
method Complete classification without symmetric assumptions, focusing on moment matrix and a new functional in dimension two.
result The moment matrix is the only SL(n) covariant valuation for n≥3, and a new functional appears in dimension two.
Novel Orlicz regrets consistently bound environmental variable statistics.
problem Consistent evaluation of stochastic environmental variables like water quality indices.
method Proposed novel Orlicz regrets for upper and lower bounds.
result Explicit linkage between Orlicz regrets and divergence risk measures.
Paper bounds Kähler manifolds' diameter using Orlicz spaces and complex Monge-Ampère equations.
problem Establishing diameter bounds for Kähler manifolds in Orlicz spaces.
method Proving a priori estimates for solutions of complex Monge-Ampère equations in Orlicz spaces using Kołodziej's and Guo-Phong-Tong-Wang's approaches.
result Uniform estimates for Green's function and its gradient for Kähler metrics.
We provide a variety of results for (quasi)convex, law-invariant functionals defined on a general Orlicz space, which extend well-known results in the setting of bounded random variables. First, we show that Delbaen's representation of convex functionals with the Fatou property, which fails in a general Orlicz space, c…
De Rham theorem extended to Orlicz cohomology.
problem Extending de Rham's theorem to a broader class of cohomology.
method Proving isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. result Isomorphism between de Rham Lφ-cohomology and simplicial ℓφ-cohomology. Study explores robust Orlicz spaces in finance, showing separability implications.
problem Understanding robustness in financial and economic contexts.
method Distinguished two constructions of robust Orlicz spaces: top-down and bottom-up.
result Separability of robust Orlicz spaces has strong implications for dominatedness and order completeness.
Generalizes Sobolev IPM for graph-based measures using Orlicz geometric structure.
problem Limitation of Le et al. (2025) framework to Lp geometry. method Generalizes Sobolev IPM through Orlicz geometric structure, employing convex functions to capture nuanced geometric relationships.
result GSI-M reduces to a simple univariate optimization problem, achieving remarkable computational efficiency.
Sharp concentration inequalities for sub-Orlicz random variables with phase transition at α=2.
problem Developing concentration inequalities for sub-Orlicz random variables with phase transition.
method New theoretical analysis framework involving variance and min/max functions of Orlicz tails.
result Sharp concentration inequalities with phase transition at α=2 for sub-Orlicz random variables.
The paper analyzes elicitability of return risk measures and their scoring functions.
problem Elicitability of return risk measures and their scoring functions.
method Dual representation results for convex and geometrically convex return risk measures, axiomatic characterizations of Orlicz premia, and construction of strictly consistent scoring functions.
result Orlicz premia are the only elicitable return risk measures under different sets of conditions.
Proposes GST for efficient computation of generalized Sobolev transport on graph metrics.
problem Optimal transport for measures on graph metric spaces with limited flexibility.
method Introduces GST, a generalized variant of Sobolev transport that adapts to various geometric structures.
result GST is significantly faster than Orlicz-Wasserstein (OW) and demonstrates advantages in document classification and topological data analysis.
We investigate a variety of stability properties of Haezendonck-Goovaerts premium principles on their natural domain, namely Orlicz spaces. We show that such principles always satisfy the Fatou property. This allows to establish a tractable dual representation without imposing any condition on the reference Orlicz func…
Paper studies convex risk measures linked to optimization.
problem Risk assessment in finance and insurance.
method Investigates a wide class of risk measures on Orlicz spaces.
result Characterizes the dual of risk measures and provides complementary representations.
In the dual LΦ∗ of a Δ2-Orlicz space LΦ, that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology τ(LΦ∗,LΦ) if and only if on each order interval [−ζ,ζ]={ξ:−ζ≤ξ≤ζ} (ζ∈LΦ∗), it is lowe…
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
Researchers prove uniqueness and continuity of solution to L_p dual Minkowski problem.
problem Proving uniqueness and continuity of solution to L_p dual Minkowski problem.
method Established new Minkowski-type inequalities related to optimization problem.
result Uniqueness and continuity of solution for general convex bodies when q<p.