A unique volume minimizer is found in a class of convex bodies.
problem Finding a unique minimizer for a reverse isoperimetric problem.
method Proving a reverse quermassintegral inequality.
result The convex hull of two balls is a unique minimizer among λ-concave bodies. Paper proves stability of quermassintegral inequalities using inverse curvature flow.
problem Stability of quermassintegral inequalities for nearly spherical sets.
method Inverse curvature flow with special rescaling to study quermassintegral inequalities.
result Decreasing rate of k-th quermassintegral is faster than Fraenkel asymmetry for nearly spherical sets.
The paper improves inequalities for nearly spherical sets using quermassintegrals.
problem Improving inequalities for nearly spherical sets.
method Establishing quantitative Alexandrov-Fenchel inequalities for quermassintegrals.
result Lower bounds on the (k,m)-isoperimetric deficit found using spherical deviation and asymmetry. 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.
Paper explores curvature flows on spheres to prove inequalities.
problem Prove inequalities for convex domains on spheres.
method Designs locally constrained curvature flows to preserve quermassintegrals.
result Flow convergence to a round sphere would settle inequalities.
The study proves stability of quermassintegral inequalities in hyperbolic space.
problem Stability of quermassintegral inequalities for horospherically convex hypersurfaces in hyperbolic space.
method Using initial value independent curvature estimates for locally constrained flows of inverse type.
result Explicit exponent of the deficit in the quermassintegral inequality is given and does not depend on dimension.
New Lp-Steiner quermassintegrals defined from Steiner formula.
problem Defining new Lp-Steiner quermassintegrals. method Analogy to classical Steiner formula, investigating properties in convex bodies.
result Rotation and reflection invariant valuations in convex bodies.
Study flows to analyze sphere quermassintegrals.
problem Analyze quermassintegrals on the sphere.
method Use two types of flows to study quermassintegrals.
result Establish Alexandrov-Fenchel inequalities for the sphere.
New convexity concept applied to sphere yields quermassintegral inequalities.
problem Proving quermassintegral inequalities for horo-convex hypersurfaces on the sphere.
method Smooth convergence of Guan/Li flow for inverse type applied to horo-convex hypersurfaces.
result Full set of quermassintegral inequalities for horo-convex hypersurfaces proved.
Paper proves a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
problem Proving a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
method Using a locally constrained nonlinear curvature flow to preserve the n-th quermassintegral and decrease the k-th quermassintegral. result Obtained the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in Bn+1. Flow preserves quermassintegrals, converging to a geodesic sphere.
problem Volume preservation issue in sphere mean curvature flow.
method Introduced a mean curvature flow with a global term to keep quermassintegrals fixed.
result Flow exists for all times and converges to a geodesic sphere.
We give a simple proof of the insoperimetric inequality for quermassintegrals of non-convex starshaped domains, using a reslut of Gerhardt \cite{G} and Urbas \cite{U} on an expanding geometric curvature flow.
The paper finds the largest eigenvalue for a specific type of domain in hyperbolic space.
problem Finding the domain with the largest first eigenvalue for a given volume and boundary conditions.
method Shape optimization for the first eigenvalue of the p-Laplace operator in hyperbolic space.
result The concentric annular region maximizes the first eigenvalue among multiply-connected domains.
The paper proves inequalities for convex hypersurfaces in spheres and hyperbolic spaces.
problem Understanding geometric properties of convex hypersurfaces in curved spaces.
method Proving identities and inequalities for closed, strictly convex hypersurfaces in spheres and hyperbolic/de Sitter space.
result Generalized Blaschke-Santaló type inequalities and quermassintegral inequalities in hyperbolic/de Sitter space.
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.
The paper proves stability of inequalities for nearly spherical sets in various spaces.
problem Stability of geometric inequalities for nearly spherical sets.
method Deriving a quantitative quermassintegral inequality and applying it to derive stability results.
result Stability of geometric inequalities involving weighted curvature integrals and quermassintegrals for nearly spherical sets in Rn+1 and Hn+1. In this article, we introduce a new type of mean curvature flow for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume is a constant and the surface area evolves monotonically. Moreover, for a bounde…
Proves inequalities for hypersurfaces in the sphere, solving a long-standing problem.
problem Proving inequalities for hypersurfaces in the sphere.
method Using mixed volumes and quermassintegrals, the authors prove inequalities equivalent to a sharp relation among three adjacent quermassintegrals.
result Proves inequalities for hypersurfaces in the sphere, equivalent to a sharp relation among three adjacent quermassintegrals.
The paper proves inequalities for hypersurfaces in a unit ball with specific boundary conditions.
problem Proving inequalities for hypersurfaces in a unit ball with capillary boundary conditions.
method Developed a curvature flow for θ-capillary hypersurfaces and used it to prove quermassintegral inequalities. result Proved full set of quermassintegral inequalities for θ-horocap-convex hypersurfaces. The article proves inequalities for capillary hypersurfaces in hyperbolic space.
problem Proving inequalities for capillary hypersurfaces in hyperbolic space.
method Constructing a new locally constrained inverse curvature flow.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space.
New weighted geometric inequalities for hypersurfaces in R^n proved.
problem Proving new weighted geometric inequalities for hypersurfaces in R^n.
method Proof of a family of sharp weighted inequalities involving weighted k-th mean curvature integral and quermassintegrals.
result Generalization and new proof of Wei and Zhou's result without relying on earlier results.
The paper proves geometric inequalities in sphere using locally constrained flows.
problem Deriving geometric inequalities in sphere.
method Established the longtime existence and convergence of a locally constrained flow.
result Proved new families of three-term geometric inequalities in sphere.
The paper proves geometric inequalities and their stabilities for curves in hyperbolic space.
problem Geometric inequalities and their stabilities for curves in hyperbolic space.
method Curve flow for shifted principal curvatures, Heintze-Karcher type inequality for h-convex curves.
result Geometric inequalities and their stabilities for curves in hyperbolic space.
In this paper we first establish an optimal Sobolev type inequality for hypersurfaces in $\H^n$(see Theorem \ref{mainthm1}). As an application we obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and quermassintegrals. Precisely, we prove a following geometric inequality in the hyperbolic space …
We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function f of the principal curvatures which is inverse concave and has dual f∗ approachi…
Paper solves inequalities for convex hypersurfaces with free boundary in a ball.
problem Finding inequalities for convex hypersurfaces with free boundary in a ball.
method Introduced quermassintegrals and used a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.
result Obtained new Alexandrov-Fenchel inequalities for convex free boundary hypersurfaces.
Paper solves anisotropic capillary Minkowski problem for p ≥ 1.
problem Anisotropic capillary convex bodies and their properties.
method Introduced anisotropic capillary p-sum and computed variations of quermassintegrals. result Solved the anisotropic capillary Lp-Minkowski problem for p≥1. 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 studies curvature flows in hyperbolic space and proves geometric inequalities.
problem Proving geometric inequalities in hyperbolic space using curvature flows.
method Locally constrained curvature flows, h-convexity, and shifted principal curvatures.
result Established new sharp geometric inequalities comparing curvature integrals to quermassintegrals.
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.
Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.
In this paper, we establish a generalised Blaschke-Santalò inequality for convex bodies in Rn+1. This inequality gives an upper bound estimate for the product of dual quermassintegrals of convex body and its polar set. Our argument is based on induction on dimensions.
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…
Study on curvature flow in Minkowski space for cocompact hypersurfaces.
problem Investigating curvature flow in Minkowski space for cocompact hypersurfaces.
method Investigation of the cocompact inverse \(σ_k\) curvature flow in Minkowski space.
result Longtime existence and convergence of the curvature flow established.
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 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 paper proves geometric inequalities for pinched convex hypersurfaces in de Sitter space.
problem Geometric inequalities for convex hypersurfaces in de Sitter space.
method Locally constrained flows with initial compact spacelike hypersurfaces pinched in de Sitter space.
result Established geometric inequalities related to quermassintegrals and weighted curvature integrals.
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.
Paper solves inequalities for capillary hypersurfaces in half-spaces.
problem Finding inequalities for convex capillary hypersurfaces in half-spaces.
method Introduced quermassintegrals and constructed a new locally constrained curvature flow to prove convergence to spherical caps.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces Σ with nonnegative sectional curvature in Hn. As an application, we prove the hyperbolic Alexandrov-Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in $\ma…
In this paper we prove the following geometric inequality in the hyperbolic space $\H^n$ (n≥5), which is a hyperbolic Alexandrov-Fenchel inequality, \[\begin{array}{rcl} \ds \int_Σ\s_4 d μ\ge \ds\vs C_{n-1}^4ω_{n-1}\left\{\left(\frac{|Σ|}{ω_{n-1}} \right)^\frac 12 + \left(\frac{|Σ|}{ω_{n-1}} \right)^{\frac 12\frac…
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.
The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.
problem Proving inequalities for convex hypersurfaces.
method Introducing a flat logarithmic centro-affine geometry and using Bochner formulas.
result Established new Poincaré and Brunn-Minkowski inequalities.
Classifies reversible and strongly reversible elements in quaternionic groups.
problem Classifying reversible and strongly reversible elements in quaternionic groups.
method Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
result Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
The paper classifies reversible and strongly reversible elements in Hermitian isometry groups.
problem Classifying reversible and strongly reversible elements in Hermitian isometry groups.
method Classification through group theory and algebraic manipulation.
result New classification of strongly reversible elements in Sp(n).
The paper classifies reversible and strongly reversible elements in quaternionic hyperbolic spaces.
problem Classifying reversible and strongly reversible elements in quaternionic hyperbolic spaces.
method Analyzing conjugacy classes and using properties of quaternionic hyperbolic spaces and their isometry groups.
result All elements of the isometry group of quaternionic hyperbolic spaces are strongly reversible.
This paper classifies reversible and strongly reversible elements in affine groups.
problem Classifying reversible and strongly reversible elements in affine groups.
method Identifying affine transformations and using conjugacy by involutions.
result Classification of reversible and strongly reversible elements in affine groups.
Study finds mean reversion strategies perform well on historical data but fail in recent market conditions.
problem Performance of mean reversion strategies in recent market data.
method Empirical investigation of three mean reversion strategies (PAMR, OLMAR, TCO) on historical S&P 500 data and benchmark datasets.
result Mean reversion strategies may fail in recent market conditions, especially with transaction costs.