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. 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 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.
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.
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.
We present a novel, log-radius profile representation for convex curves and define a new operation for combining the shape features of curves. Unlike the standard, angle profile-based methods, this operation accurately combines the shape features in a visually intuitive manner. This method have implications in shape an…
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.
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.
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.
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.
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.
Active-set algorithm improves Cox regression for shape-restricted covariates.
problem Improving Cox regression for shape-restricted covariates.
method Shape-restricted inference using active-set optimization for spline basis expansion.
result Active-set algorithm produces accurate linear covariate effect estimates.
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 solves a thermodynamics problem about crystal shape.
problem Understanding if minimizing free energy with convex potential and mass constraint generates a convex crystal.
method Utilized a stability theorem, convexity, and a new maximum principle approach to prove a three-dimensional convexity theorem.
result Completely settled the Almgren problem in R3 under generic conditions. Paper tackles multivariate shape-constrained convex regression problems.
problem Fitting a convex function to data with component-wise monotonicity and uniform Lipschitz continuity.
method Least squares estimator via solving a constrained convex quadratic programming problem. Efficient algorithms designed: sGS-ADMM and pALM.
result Both proposed algorithms outperform state-of-the-art methods in numerical experiments.
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.
New Heintze-Karcher inequality helps understand droplet shapes.
problem Characterize the shape of droplets inside smooth containers.
method Obtained a new form of the Heintze-Karcher inequality for mean convex hypersurfaces with boundary on curved substrates.
result New mathematical inequality aids in understanding droplet shapes.
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.
Kernel-based tests for shape constraints in finance.
problem Enforcing shape relations on latent functions in financial econometrics.
method Kernel-based nonparametric framework for mean-variance optimization.
result Established statistical properties and a joint Wald-type statistic for testing shape constraints.
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…
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 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.
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 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. New framework classifies high-dimensional shapes using ray intersections, establishing data requirements.
problem Classifying high-dimensional shapes in real-world data.
method Ray-based classification (RBC) framework using intersections of one-dimensional representations (rays) with shape boundaries.
result Established bounds on the number of rays necessary for shape classification, defined by key angular metrics.
One-dimensional crystals have convex shapes under certain conditions.
problem Determining if one-dimensional crystals have convex shapes.
method Analyzing the free energy under mass constraints and convexity assumptions.
result In one dimension, crystals have convex shapes under given conditions.
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.
We consider convex symmetric lens-shaped networks in R^2 that evolve under curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks converge to a self-similarly shrinking network, which we prove to be unique in an ap…
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.
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.
Paper tackles hard shape constraints in kernel machines.
problem Enforcing shape requirements in a hard fashion is challenging.
method Tightened second-order cone constrained reformulation for kernel machines.
result Performance guarantees and efficiency demonstrated in various applications.
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.
Paper introduces length measures for curves and convex shapes, proving isoperimetric and distance properties.
problem Characterizing and comparing convex shapes using length measures.
method Developed length measures for curves and convex shapes, derived properties, and introduced a new distance metric.
result Unique convex curve maximizes signed area among curves with same length measure.
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.
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.
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.
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.
Stable nets on convex hypersurfaces maintain their shape under small perturbations.
problem Maintaining the shape of nets on convex surfaces under slight changes.
method Constructing stable geodesic nets on convex hypersurfaces.
result Stable geodesic nets on convex hypersurfaces do not change shape under small perturbations.
In this study, we extend the optimal execution problem with convex market impact function studied in Kato (2014) to the case where the market impact function is S-shaped, that is, concave on [0,xˉ0] and convex on [xˉ0,∞) for some xˉ0≥0. We study the corresponding Hamilton-Jacobi-…
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.
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.
Unified framework for hard affine SDP constraints in vRKHSs.
problem Incorporating shape constraints into predictive models for rich function classes.
method Unified convex optimization framework using second-order cone tightening.
result Unified and modular approach for handling multiple shape constraints.
In his paper "Shapes of Polyhedra and Triangulations of the Sphere", Thurston found that the set of shapes of convex polyhedra with prescribed cone-deficits has a complex hyperbolic structure. Inspired by his work, this paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits. We s…
Gradient flow expands curves to round shapes.
problem Expanding curves to round shapes.
method Steepest descent L2-gradient flow of entropy.
result Flow converges to a round expanding circle for various initial curves.
We consider a compact, star-shaped, mean convex hypersurface Σ2⊂R3. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which …
Functional BART adds shape priors to Bayesian tree regression for better curve fitting.
problem Regression with function-on-scalar data and shape constraints.
method Bayesian tree structure with spline representations, customized Bayesian backfitting algorithm, shape priors.
result Improved estimation and prediction accuracy with shape priors.
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.
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.