The paper proves stability of Wulff shapes using anisotropic curvature functionals.
problem Stability of Wulff shapes under anisotropic curvature.
method Estimates distance to Wulff shape using Lp-norm of traceless F-Hessian of a foliating function. result Quantitative stability results for anisotropic inequalities and problems.
Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.
problem Proving anisotropic extrinsic radius pinching inequality for hypersurfaces.
method Analyzes anisotropic mean curvatures and studies equality cases.
result Almost extremal hypersurfaces are close to Wulff shape.
Local minimizers are convex and close to Wulff shapes.
problem Finding local minimizers in anisotropic isoperimetric problems.
method Showed local minimizers are geodesically convex and small smooth perturbations of tangent Wulff shapes.
result Local minimizers are quantitatively close to Wulff shapes.
Simplifies Wulff theorem for crystalline shapes using Minkowski Theory.
problem Proving the Wulff theorem for crystalline integrands.
method Direct approach using Minkowski Theory to exploit convex properties.
result Simpler proof of the Wulff theorem for crystalline shapes.
New anisotropic surfaces can have non-Wulff shapes without being Wulff shapes.
problem Existence of non-Wulff anisotropic surfaces.
method Analysis of anisotropic surface energy and mean curvature flow.
result Non-uniqueness of anisotropic surfaces with constant mean curvature.
In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class C1. Moreover, applications of this result are given.
New theorem proves convex bodies with specific curvature measures are rescaled Wulff shapes.
problem Characterizing convex bodies based on anisotropic curvature measures.
method Analyzing k-th anisotropic curvature measures and their relation to anisotropic perimeter.
result Arbitrary convex bodies with specific curvature measures are rescaled Wulff shapes.
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. We show that for elliptic parametric functionals whose Wulff shape is smooth and has strictly positive curvature, any surface with constant anisotropic mean curvature which is a topological sphere is a rescaling of the Wulff shape.
Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide t…
Uniqueness of stable, non-smooth hypersurfaces with constant anisotropic mean curvature.
problem Identifying stable, non-smooth hypersurfaces with constant anisotropic mean curvature.
method Study of piecewise-smooth hypersurfaces with anisotropic energy, proving uniqueness of the Wulff shape under certain conditions.
result Closed stable equilibrium hypersurfaces are unique and the Wulff shape when the anisotropic energy density is twice continuously differentiable and convex.
We study the stability of closed, not necessarily smooth, equilibrium surfaces of an anisotropic surface energy for which the Wulff shape is not necessarily smooth. We show that if the Cahn Hoffman field can be extended continuously to the whole surface and if the surface is stable, then the surface is, up to rescaling…
Optimal inequality for free boundary hypersurfaces in convex domains.
problem Proving an optimal Heintze-Karcher inequality for free boundary hypersurfaces.
method Analyzing anisotropic free boundary hypersurfaces in convex domains.
result Optimal Heintze-Karcher-type inequality achieved for anisotropic free boundary Wulff shapes.
The paper introduces a new flow to converge to a Wulff shape from a smooth convex hypersurface.
problem Proving Alexandrov-Fenchel inequalities for anisotropic mixed volumes.
method Introducing a fully nonlinear locally constrained anisotropic curvature flow.
result The flow converges smoothly and exponentially to a scaled Wulff shape.
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.
Given a positive function F on S n satisfying an appropriate con-vexity assumption, we consider hypersurfaces for which a linear combination of some higher order anisotropic curvatures is constant. We define the varia-tional problem for which these hypersurfaces are critical points and we prove that, up to translations…
Study finds unique critical points for anisotropic surface energy.
problem Finding unique critical points for anisotropic surface energy.
method Proving finite unions of disjoint open Wulff shapes are volume-constrained critical points.
result Finite unions of disjoint open Wulff shapes are the only critical points.
Study shows limits of volume-constrained sets are finite unions of Wulff shapes.
problem Analyzing the behavior of sets with degenerating ellipticity.
method Proving rigidity of L1-accumulation points of volume-constrained almost-critical sets. result Limits of volume-constrained sets are finite unions of φ-Wulff shapes. The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.
problem Anisotropic length preservation in curve deformation.
method A curve flow that maintains anisotropic length, analyzed for convex closed curves.
result Convex curves evolve to homothetic limits of Wulff shapes as time approaches infinity.
Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
Study on surfaces with constant anisotropic mean curvature in 3D space.
problem Characterizing surfaces with constant anisotropic mean curvature.
method Analyzing uniformly elliptic anisotropic functionals and proving properties of surfaces.
result Characterization of surfaces with constant anisotropic mean curvature.
Given a positive function F on Sn which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature Mr for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature Hr. We get integral formulas of Minkowski type for compact hypersurfaces in Rn+1. We give some new characteriz…
We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider n≥2, p∈(1,+∞) and Σ an n-dimensional, closed hypersurface in Rn+1, boundary of a convex, open set. We show that …
Compact hypersurfaces minimize area in convex cones with free boundary.
problem Finding compact hypersurfaces minimizing area in convex cones with free boundary.
method Minimizing an anisotropic area functional under a volume constraint.
result Compact hypersurfaces are contained in a Wulff-shape.
The paper studies curvature measures and volume-preserving flows on convex bodies.
problem Characterizing and understanding convex bodies through anisotropic curvature measures.
method Developed anisotropic curvature measures, used Minkowski formulas and Heintze-Karcher inequalities, and analyzed volume-preserving flows.
result Characterized Wulff shapes via anisotropic curvature measures and proved convergence of volume-preserving flows.
Paper extends Wente's result to anisotropic capillary surfaces in half-spaces.
problem Extending Wente's result to anisotropic capillary surfaces.
method New Heintze-Karcher inequality and Minkowski formula.
result Anisotropic capillary hypersurfaces in half-spaces are Wulff shapes.
Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.
problem Understanding the asymptotic behavior of anisotropic mean curvature flow.
method Established local gradient estimates for anisotropic p-harmonic functions and weak solutions of IAMCF. result Weak IAMCF is asymptotic to the expanding Wulff shape solution at infinity.
New curvature measures characterize non-convex Wulff shapes in normed spaces.
problem Characterizing non-convex sets with curvature measures.
method Extending curvature measures to non-convex and non-smooth sets in normed spaces.
result Finite unions of disjoint Wulff shapes are the only sets with proportional curvature measures.
In this paper, we show that the inverse anisotropic mean curvature flow in Rn+1, initiating from a star-shaped, strictly F-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the C∞ topology. As an application, we p…
We study surfaces with constant anisotropic mean curvature which are invariant under a helicoidal motion. For functionals with axially symmetric Wulff shapes, we generalize the recently developed twizzler representation of Perdomo to the anisotropic case and show how all helicoidal constant anisotropic mean curvature s…
We study the motion of discrete interfaces driven by ferromagnetic interactions on the two-dimensional triangular lattice by coupling the Almgren, Taylor and Wang minimizing movements approach and a discrete-to-continuum analysis, as introduced by Braides, Gelli and Novaga in the pioneering case of the square lattice. …
We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a …
We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of Rn. Our result applies to…
The paper proves a Wulff inequality for minimal submanifolds with boundary in Euclidean space.
problem Proving a Wulff inequality for minimal submanifolds with boundary.
method Associating a nonnegative anisotropic weight to the boundary of minimal submanifolds and proving the inequality.
result The Wulff inequality constant is independent of the weights and depends only on m and n. New proof of Wulff-Gage inequality with applications.
problem Proving the Wulff-Gage isoperimetric inequality.
method Provided a new proof of the inequality for origin-symmetric convex bodies.
result Uniqueness of log-Minkowski problem and new proof of log-Minkowski inequality.
Paper studies stability of curved surfaces in a half-space.
problem Stability of anisotropic capillary hypersurfaces in a half-space.
method Analyzes weak stability and proves Bernstein-type theorems.
result Compact hypersurfaces are stable if and only if they are a truncated Wulff shape.
Given a positive function F on Sn which satisfies a convexity condition, we define the r-th anisotropic mean curvature function HrF for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature function. Let X:M→Rn+1 be an n-dimensional closed hypersu…
Given a positive function F on Sn which satisfies a convexity condition, for 1≤r≤n, we define the r-th anisotropic mean curvature function HrF for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature function. We prove that a compact embedded hypersurface…
We prove that, for every closed (not necessarily convex) hypersurface Σ in Rn+1 and every p>n, the Lp-norm of the trace-free part of the anisotropic second fundamental form controls from above the W2,p-closeness of Σ to the Wulff shape. In the isotropic setting, we provide a simpler proof. T…
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. Novel framework improves randomized smoothing for various norms.
problem Developing robust defenses against adversarial attacks.
method Proposed a novel framework for devising and analyzing randomized smoothing schemes.
result Significantly improved certified accuracy in ℓ1 on standard datasets.
Study on shapes in Heisenberg group with convex body norms.
problem Characterize shapes in the Heisenberg group H1 induced by convex bodies. method Compute perimeter variation, define mean curvature, and analyze foliations.
result Existence of constant mean curvature spheres in the Heisenberg group.
Generalizes Roe's theorem to noncompact hypersurfaces.
problem Obtaining index theorems for noncompact manifolds.
method Extending Roe's partitioned manifold index theorem to noncompact hypersurfaces.
result Equality between two classes in the K-theory of the Roe algebra of N.
Geomstats introduces shape module for analyzing shapes of objects.
problem Analyzing shapes of objects represented as landmarks, curves, and surfaces.
method Implementing shape spaces, group actions, fiber bundles, quotient spaces, and Riemannian metrics.
result Users can compare, average, and interpolate shapes inside shape spaces.
A new shape space allows optimization of non-smooth shapes in fluid mechanics.
problem Optimizing non-smooth shapes in fluid mechanics.
method Constructing a product manifold to include piecewise-smooth shapes.
result Numerical results show applicability in minimizing viscous energy dissipation.
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.
Optimizes shapes in uncertain Navier-Stokes flow problems.
problem Optimizing shapes with geometric constraints and physical uncertainty.
method Multi-shape calculus and stochastic augmented Lagrangian method.
result Successfully optimized shapes in uncertain Navier-Stokes flow.
New method reconstructs 3D shapes from 2D images using Kendall's shape space.
problem Reconstruct 3D shapes from 2D images, especially for rare specimens.
method Kendall's shape space approach with prior information.
result More robust and plausible shapes compared to previous methods.