The paper solves curvature measure problem in hyperbolic space.
problem Prescribed curvature measure problem in hyperbolic space.
method Establishing C^2 regularity estimates for solutions to fully nonlinear PDE.
result Existence of star-shaped k-convex bodies with prescribed curvature measures.
We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped (n−k)-convex bodies with prescribed k-th curvature measures (k>0) has been a longstanding problem. This is settled in this paper through the establishment of a crucial C2 a priori estimate for the c…
The study proves the existence of k-convex hypersurfaces for specific curvature equations.
problem Proving the existence of k-convex hypersurfaces for Hessian curvature equations. method Combining a priori estimates with the continuity method, and establishing a constant rank theorem.
result Existence and uniqueness of k-convex hypersurfaces for both nonhomogeneous and homogeneous Hessian curvature equations. Study eigenvalues for special curvature equations on star-shaped surfaces.
problem Eigenvalue problem for prescribed curvature equations on star-shaped, k-convex hypersurfaces.
method Established existence of a unique eigenvalue and hypersurface through uniform estimates in p for Lp-type equations.
result Existence of a unique eigenvalue and associated hypersurface under certain conditions.
We study relations of some classes of k-convex, k-visible bodies in Euclidean spaces. We introduce and study \textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular, \textrm{k-circular convex} and \textrm{k-circular visible} ones. Investigati…
The paper studies curvature flows of star-shaped hypersurfaces and proves convergence to spheres.
problem Analyzing the convergence of a class of anisotropic curvature flows.
method Using new auxiliary functions, the paper studies a class of flows with specific speed and proves convergence under certain conditions.
result The k-convex solution to the flow converges smoothly to a sphere after normalization for specific values of k, α, and β. The paper studies a curvature flow on hypersurfaces in R^(n+1).
problem Analyzing the long-term behavior of a specific type of curvature flow.
method Examining a flow defined by a non-homogeneous anisotropic speed function.
result The flow converges to a sphere for star-shaped and k-convex initial hypersurfaces.
Efficiently samples arbitrary compact bodies with polynomial complexity.
problem Uniform sampling from arbitrary compact bodies efficiently.
method Warm start algorithm under isoperimetry and volume growth condition.
result Substantial generalization of known results for convex and star-shaped bodies.
Efficient algorithm for sampling from arbitrary compact bodies.
problem Sampling from arbitrary compact bodies efficiently.
method Warm start algorithm with polynomial complexity.
result Substantial generalization of known results for convex and star-shaped bodies.
Paper solves overdetermined k-Hessian equation in exterior domains.
problem Overdetermined problem for k-Hessian equation in exterior domains. method Combining integral identities and geometric inequalities, derived general monotone formulas.
result Established general monotone formulas for k-admissible solutions. This study explores star-shaped regularizers learned from critic-based losses.
problem Understanding the structure of regularizers learned from critic-based losses.
method Optimizing critic-based loss functions over star-shaped regularizers.
result Derives exact expressions for optimal regularizers in certain cases.
We obtain estimates on both size and dimensions of the singular set at the first blow-up time of the mean curvature flow of hypersurfaces whose initial data is σk-convex.
Introduces Star-Shaped deviation measures for risk analysis.
problem Risk measurement and analysis in finance.
method Characterizes Star-Shaped deviation measures through acceptance sets and convex deviation measures.
result Exposes the relationship between Star-Shaped risk measures and deviation measures.
The paper studies curvature flows in Euclidean and hyperbolic spaces, proving smooth convergence to spheres.
problem Analyzing curvature flows in Euclidean and hyperbolic spaces.
method Introduced a class of expanding flows with specific speed functions and proved their longtime existence and smooth convergence.
result The flows converge smoothly to spheres in Euclidean and hyperbolic spaces under certain conditions.
The paper derives new inequalities for non-convex domains and flows.
problem Inequalities for non-convex domains and flows.
method Inverse curvature flow and Alexandrov-Fenchel-type inequalities.
result New inequalities for non-convex domains and flows.
New star-shaped acceptability indexes generalize existing methods.
problem Generalizing existing acceptability measures.
method Characterizing acceptability indexes through star-shaped risk measures and sets.
result Introducing concrete examples linked to various financial measures.
New set-valued star-shaped risk measures introduced for better risk assessment.
problem Improving risk assessment in financial contexts.
method Developed new set-valued star-shaped risk measures and proved their representation theorems.
result Set-valued star-shaped risk measures can be represented as unions of set-valued convex risk measures.
Flow turns star-shaped curves into circles.
problem Transforming star-shaped curves into circles.
method Gage's area-preserving flow.
result Curves evolve into circles over time.
Paper proves conjecture about star-shaped curves evolving under GAPF, but not always preserves star shape.
problem What conditions guarantee global existence of Gage's area-preserving flow for nonconvex initial curves?
method Using Dittberner's singularity analysis theory, constructed a ``flying wing'' curve to show limitations.
result Gage's area-preserving flow does not always preserve star-shapedness of evolving curves.
The paper studies dynamic star-shaped risk measures and their representation.
problem Representing dynamic star-shaped risk measures and their properties.
method Representation theorems for dynamic monetary and star-shaped risk measures.
result Dynamic star-shaped risk measures can be represented as the lower envelope of a family of dynamic convex risk measures.
The paper characterizes dynamic return and star-shaped risk measures via BSDEs.
problem Characterizing dynamic return and star-shaped risk measures.
method Characterization of star-shaped functionals and BSDEs.
result Existence of convex BSDEs with non-empty set of supersolutions.
We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K-convexity of the m-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K-convexity of the weig…
The paper finds at least four prime closed characteristics on star-shaped hypersurfaces in 8D space.
problem Finding prime closed characteristics on compact star-shaped hypersurfaces in 8D space.
method Proved existence of at least four prime closed characteristics for non-degenerate C3 compact star-shaped hypersurfaces in R8 without prime closed characteristics of Maslov-type index -1. result Existence of at least four prime closed characteristics on compact star-shaped hypersurfaces in R8. The paper proves inequalities for star-shaped and F-mean convex hypersurfaces in Rn+1.
problem Proving geometric inequalities for specific types of hypersurfaces.
method Using anisotropic p-momentum, perimeter, and volume, the paper derives inequalities for star-shaped and F-mean convex hypersurfaces. result The Wulff shape of F is the unique minimizer of the corresponding functionals among all star-shaped and F-mean convex sets. The paper characterizes law-invariant star-shaped risk measures.
problem Understanding and characterizing law-invariant star-shaped risk measures.
method Developed characterizations for positively homogeneous and star-shaped functionals, derived Kusuoka-type representations, and offered representations of general law-invariant star-shaped functionals.
result Characterizations of law-invariant star-shaped functionals, including their connections to Value-at-Risk and Expected Shortfall.
Introduces Star-Shaped DDPMs for non-Gaussian distributions.
problem Difficulties in defining DDPMs for non-Gaussian distributions.
method Star-shaped diffusion process, duality with specific Markovian diffusions, efficient algorithms.
result SS-DDPMs can model distributions like Beta, von Mises-Fisher, Dirichlet, Wishart.
The paper generalizes convex and star-shaped concepts to symplectic spaces and studies variational problems.
problem Generalizing convex and star-shaped concepts to symplectic vector spaces.
method Study of variational problems for symplectically convex and star-shaped curves.
result Extremal points of the variational problem are rigid multiply traversed conics for a range of parameters.
The paper establishes Pogorelov type estimates for solutions to Hessian quotient equations in hyperbolic space.
problem Estimating solutions to Hessian quotient equations in Lorentz-Minkowski space.
method Using a priori estimates, the paper establishes Pogorelov type estimates for k-convex solutions.
result Pogorelov type estimates for k-convex solutions to Hessian quotient equations in hyperbolic space.
In 1998 Smoczyk [Smo98] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in R3. We prove in this paper that this is true for the mean curvature flow of star-shaped hypersurfaces in Rn+1 in arbitrary …
Paper characterizes star-shaped risk measures and their properties.
problem Characterizing risk measures in the presence of liquidity risk and competitive delegation.
method Characterization of star-shaped risk measures, study of their properties.
result Star-shaped risk measures include all practically used risk measures.
This paper connects monetary and star-shaped risk measures by showing their equivalence under certain conditions.
problem Understanding the relationship between monetary and star-shaped risk measures.
method Analyzing the acceptability of 0 and the normalization property.
result Monetary risk measures are only a translation away from star-shapedness under mild conditions.
Study anisotropic flow for capillary hypersurfaces, proving new inequalities.
problem Anisotropic capillary hypersurfaces and their properties.
method Anisotropic volume-preserving mean curvature flow, new approach for strictly convex initial hypersurfaces.
result Established new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces.
Convexity preserved in curved surfaces moving at concave speeds.
problem Deforming convex surfaces with concave speeds.
method Nonlinear geometric flows with concave speed functions.
result High curvature regions remain approximately convex.
Proves a Minkowski inequality for star-shaped hypersurfaces in warped cylinders.
problem Proving a Minkowski inequality for specific types of hypersurfaces.
method Using weakly mean convex and star-shaped hypersurfaces in warped cylinders, and applying the inverse mean curvature flow.
result Sharp inequality holds for outward minimizing hypersurfaces in Schwarzschild and hyperbolic spaces.
Inverse curvature flows shape star-shaped hypersurfaces into spheres.
problem Evolution of star-shaped hypersurfaces inside a convex cone.
method Inverse curvature flows, convexity of the cone, gradient and Hölder estimates.
result Hypersurfaces converge to a round sphere as time goes to infinity.
Constructs Koszul dual algebras for star-shaped diagrams in 3-manifolds.
problem Constructing algebraic structures for 3-manifold homology.
method Uses graphical calculus to construct Koszul dual weighted A∞-algebras and dualizing bimodules. result Proves duality of constructed algebras and bimodules.
Proves optimal isoperimetric inequality in de Sitter space.
problem Optimal isoperimetric inequality for specific hypersurfaces in de Sitter space.
method Analyzes spacelike, compact, star-shaped, and 2-convex hypersurfaces in de Sitter space.
result Proves an optimal isoperimetric inequality for the specified hypersurfaces.
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.
Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.
In this paper we consider a star-shaped hypersurface flow by mean curvature. Without any assumption on the convexity, we give a new proof of gradient estimate for a short time. As an application, we also give a lower bound for the blowing up time.
This paper continues the study of Alexandrov-Fenchel inequalities for quermassintegrals for k-convex domains. It focuses on the application to the Michael-Simon type inequalities for k-curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a do…
This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev 10:368−370, 1968) on a star-shaped bounded domain in R2. Let Ω be a star-shaped bounded domain in a hypersurface of revolution, having smooth boundary. In this article, we obtain a sharp lower bound for all Steklov eigenv…
We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution …
The paper explores risk measures and arbitrage in financial markets.
problem Quantifying and managing risk in financial markets.
method Introduces new risk measure axioms and characterizes arbitrage conditions.
result Derives the consistent price interval for financial contracts.
The paper proves new inequalities in hyperbolic space using Euclidean methods.
problem Proving weighted isoperimetric inequalities in hyperbolic space.
method Using isoperimetric inequality with log-convex density in Euclidean space.
result Removed horo-convex assumption and proved new inequalities for star-shaped domains.
Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
The paper maps two types of hyperkähler manifolds and identifies their symplectic structures.
problem Mapping and identifying symplectic structures of two types of hyperkähler manifolds.
method Produced a map from star-shaped quiver varieties to Higgs bundle moduli spaces, verified stability, and showed it is a homeomorphism.
result Identified natural holomorphic symplectic structures on the two spaces.
In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic hyperbolic space. We p…