Derives stability for curvature measure near constant density, proving dual Minkowski problem solutions.
problem Stability of curvature measure near constant density
method Derives stability result for curvature measure, proves existence and uniqueness of solutions to dual Minkowski problem.
result Existence and uniqueness of solutions to dual Minkowski problem for positive indices, stability result for curvature measure.
Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of con…
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. 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.
Solves a generalized dual Minkowski problem for specific values of q.
problem Finding solutions to the generalized dual Minkowski problem for given q and star bodies.
method Variational methods
result Existence of solutions for q<0 and 0≤q≤1, sufficient condition for q>1.
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.
We introduce the dual isoperimetrix which solves the isoperimetric problem in the dual Brunn-Minkowski theory. We then show how the dual isoperimetrix is related to the isoperimetrix from the Brunn-Minkowski theory.
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…
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…
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.
Characterizes sample complexity for outcome indistinguishability in machine learning.
problem Outcome indistinguishability in machine learning, focusing on distinguishers and predictors.
method Sample complexity characterized by metric entropy of predictor and distinguisher classes, using dual Minkowski norms.
result Equivalence and tightness of sample complexity characterizations in distribution-specific and distribution-free settings.
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…
The paper classifies curves in dual affine and Lorentz-Minkowski planes with constant curvature.
problem Classifying curves with constant curvature in dual affine and Lorentz-Minkowski planes.
method Investigation of invariants under equiaffine transformations and explicit equations for curves with constant curvature.
result Curves with constant curvature in dual affine and Lorentz-Minkowski planes are classified.
Paper estimates diameter for Minkowski problem solutions.
problem Estimating diameter of solutions to Minkowski problem.
method Uniform diameter estimate for Lp dual Minkowski problem. result Uniform diameter estimate for planar Lp dual Minkowski problem. 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.
Paper solves dual Minkowski problem in 2D plane for specific curvature cases.
problem Finding the number of solutions to the dual Minkowski problem in 2D with constant curvature.
method Combining theoretical analysis and numerical estimation of an integral with parameters.
result Found the number of solutions for the constant dual curvature case when 0<q≤4. 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.
General (α,β) norms are an important class of Minkowski norms which contains the original (α,β) norms. In this note, by studying the behavior of the Darboux curves of the indicatrix, we give a characterization of 3-dimensional general (α,β) norms. By studying the isoperimetric properties of the indicatrix, as …
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 …
New Brownian motion defined in Minkowski normed spaces.
problem Constructing Brownian motion in non-Euclidean spaces.
method Singular McKean--Vlasov stochastic differential equation.
result Pathwise uniqueness of solutions to the stochastic differential equation.
Unified Minkowski problem discussed for (p,q)-mixed quermassintegrals.
problem Unified Minkowski problem for (p,q)-mixed quermassintegrals.
method Introducing (p,q)-mixed quermassintegrals and (p,q)-dual mixed curvature measure to study the Minkowski problem.
result Derivation of important properties and geometric inequalities for (p,q)-mixed quermassintegrals.
The paper proves conjectures about Minkowski norms with specific symmetry groups.
problem Proving conjectures about Minkowski norms with certain symmetries.
method Analyzing isometries of the Hessian metric for Minkowski norms invariant under SO(k)imesSO(n−k). result Proves Laugwitz and Landsberg Unicorn conjectures for Minkowski norms with the specified symmetry.
We introduce and study deformation Tb,φ of Minkowski norms in Rn, determined by a set b=(β1,…,βp) of linearly independent 1-forms and a smooth positive function φ of p variables. In particular, the Tb,φ-image of a Euclidean norm α is a Minkowski norm, whose indicat…
Study confirms the uniqueness of the unit sphere for a specific geometric problem.
problem Uniqueness of solutions to the isotropic Lp dual Minkowski problem. method Proof by contradiction and analysis of the given equation.
result The unit sphere is the only smooth, strictly convex solution.
Paper studies inverse curvature flows and solves related geometric problems.
problem Inverse curvature flows and related geometric problems.
method Analyzes a class of expanding flows with specific speeds and proves existence and convergence.
result Proves the existence and convergence of flows under certain conditions, leading to new solutions to geometric problems.
The study explores special surfaces in a normed space.
problem Constant Gaussian and mean curvature surfaces in normed spaces.
method Analyzes rotational surfaces with specific curvature properties.
result Generalizes catenoid, pseudo-sphere, and Delaunay surfaces.
The paper solves a specific Minkowski problem for capillary hypersurfaces.
problem Finding capillary convex bodies with prescribed dual curvature measures.
method Reduction to a Monge-Ampère type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution for θ∈(0,2π). The paper studies Minkowski norms and Hessian isometries induced by isoparametric foliations on spheres.
problem Understanding Minkowski norms and Hessian isometries induced by isoparametric foliations.
method Constructing Minkowski norms using spherical coordinates, studying Hessian isometries using spherical local frames, and proving properties of these isometries.
result Proves the existence and properties of Hessian isometries induced by isoparametric foliations on spheres.
We prove an analogue of the classical Steiner formula for the Lp affine surface area of a Minkowski outer parallel body for any real parameters p. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Stein…
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 dual Lp-Minkowski problem with p<0<q is investigated in this paper. By proving a new existence result of solutions and constructing an example, we obtain the non-uniqueness of solutions to this problem.
This paper classifies solutions for a specific geometric problem.
problem Classifying solutions for the planar isotropic Lp dual Minkowski problem. method Converted the ODE for the solution into an integral and studied its asymptotic behavior, duality, and monotonicity.
result Complete classification of solutions for the equation.
Formula proves monotonicity for anisotropic minimal hypersurfaces.
problem Understanding anisotropic minimal hypersurfaces.
method Proved a monotonicity formula under a sign assumption on the Minkowski norm.
result Monotonicity formula for anisotropic minimal hypersurfaces.
Study anisotropic capillary surfaces in a wedge using generalized Minkowski norms.
problem Understanding capillary surfaces with anisotropic forces.
method Generalized Minkowski norm on the unit sphere, new Minkowski formulae, Heintze-Karcher inequality.
result Proved an Alexandrov-type theorem in the anisotropic setting.
Paper proves smoothness of solutions to a complex geometric problem.
problem Smoothness of solutions to the degenerate Lp Dual Minkowski problem. method Inspired by Guan and Li's approach for the Aleksandrov problem, the authors derive C1,1 estimates. result Proves solutions are C1,1 regular. Paper solves a new Minkowski problem for a specific type of rigidity.
problem Solving a new Minkowski problem for a specific type of rigidity.
method Developed a nonlinear partial differential equation and used a curvature flow method.
result Existence of smooth non-even solutions to the p-th dual Minkowski problem for p < n-2.
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…
The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the q-th dual curvature measure of an origin-symmetric convex body in Rn. A full solution to this is given when 1<q<n. The necessary and suffic…
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.
Paper solves a geometric problem involving mixtures of area and curvature measures.
problem Investigates a geometric problem involving mixtures of area and curvature measures.
method Establishes a gradient estimate to prove the existence of a solution.
result Proves the existence of an even, smooth, strictly convex solution for 1<p<q≤k+1. Article provides polytopes as dual unit balls of Thurston norms on 3-manifolds.
problem Understanding the dual unit ball shape of Thurston norms.
method Introduced a family of polytopes in Z^2g that can be dual unit balls of Thurston norms on 3-manifolds.
result Polytopes with mod 2 congruent vertices can be realized as dual unit balls of Thurston norms.
Proposes a new K-means method for efficient clustering of nonlinear data.
problem Challenges of kernel K-means, including high memory usage and computational inefficiency.
method Combines linear and nonlinear approaches using explicit feature maps based on spectral analysis.
result Demonstrates Explicit Kernel Minkowski Weighted K-means (Explicit KMWK-means) reduces memory usage and improves efficiency.
Study complete 3D λ-translators in Minkowski space with constant properties.
problem Classify 3D space-like λ-translators with specific constant properties.
method Obtained classification theorem through analysis of constant norm and f4. result Classification theorem for 3D complete space-like λ-translators.
The paper proves a Minkowski-like theorem for tetrahedra in dS3 and AdS3.
problem Formulating and proving a constant-curvature, holonomy-valued Lorentzian analogue of Minkowski theorem for tetrahedra.
method Formulated and proved a Lorentzian analogue of Minkowski theorem for tetrahedra in dS3 and AdS3.
result A unique strictly convex tetrahedron can be reconstructed from four non-trivial based SO+(1,2) holonomies.
Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of resu…
The traditional Minkowski distances are induced by the corresponding Minkowski norms in real-valued vector spaces. In this work, we propose novel statistical symmetric distances based on the Minkowski's inequality for probability densities belonging to Lebesgue spaces. These statistical Minkowski distances admit closed…
Constructs a convex Finsler metric on vector bundles under specific conditions.
problem Creating a convex Finsler metric on vector bundles with positive curvature.
method Uses the negativity of direct image bundles and Minkowski inequality for norms.
result Shows how to upgrade a Kobayashi positive Finsler metric to a convex one.
We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifold…