New formulas for geometric measures in vector spaces.
problem Local additive kinematic formulas for vector spaces.
method Introducing dual area measures and proving their convolution product.
result Local additive kinematic formulas in hermitian vector spaces.
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. New Steiner formula for Lp affine surface area in Minkowski theory.
problem Developing a new Steiner formula for Lp affine surface area. method Proving a new Steiner formula for the Lp affine surface area of a Minkowski outer parallel body. result New curvature measures with properties not previously seen in literature.
New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.
problem Characterizing properties of ball-convex shapes.
method Introducing illumination bodies and weighted illumination bodies, proving convexity, and establishing surface area measures.
result Illumination bodies are convex and provide surface area measures for ball-convex shapes.
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…
A simpler edge-based discretization method without dual volumes.
problem Efficiently computing edge-based discretization vectors without forming dual volumes.
method Directly compute edge-midpoint vectors and reduce dual volume formation.
result Significant reduction in computing time for tetrahedral grids.
The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.
problem Understanding ergodicity of geodesic flows on infinite Riemann surfaces.
method Analyzing random walks on the dual graph of pants decompositions.
result Equivalence between ergodicity of geodesic flows and recurrence of random walks.
Study spherical convex bodies using Lp-floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced Lp-floating areas and curvature entropy for spherical convex bodies. result Established isoperimetric inequalities and dual isoperimetric inequalities.
Investigates dual foliations of polygon spaces based on area and perimeter.
problem Understanding dual foliations of polygon spaces guided by area and perimeter.
method Investigated topology of leaves, determined homology groups, and extended isoperimetric duality.
result Homology groups and homotopy types of polygon spaces are determined.
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.
Solves a problem related to dual Orlicz curvature measure.
problem Solving the dual Orlicz-Minkowski problem.
method Introduced dual Orlicz curvature measure and established a variational formula.
result Provided a solution to the dual Orlicz-Minkowski problem.
Dual representations for robust risk measures and uncertainty sets.
problem Characterizing continuity of robust risk measures and their uncertainty sets.
method Develop dual representations for robust risk measures and uncertainty sets based on distinct geometric assumptions.
result Two dual frameworks for consolidated uncertainty sets are complementary, not interchangeable.
Study on Santaló point for convex bodies in normed spaces.
problem Exploring Santaló point for convex bodies in normed spaces.
method Existence and uniqueness proof for C1 norms, dual Santaló point for smooth curved unit balls. result Existence and uniqueness of Santaló point for convex bodies in normed spaces.
In order to evaluate the quality of the scientific research, we introduce a new family of scientific performance measures, called Scientific Research Measures (SRM). Our proposal originates from the more recent developments in the theory of risk measures and is an attempt to resolve the many problems of the existing bi…
New chiral minimal surfaces derived from quartz network.
problem Finding new triply-periodic minimal surfaces.
method Using dual graphs of quartz and its dual, generating area-minimizing meshes, and identifying flat point structures.
result Identified a new family of chiral triply-periodic minimal surfaces.
New convex body defined to prove Busemann Random Simplex Inequality.
problem Proving Busemann Random Simplex Inequality for general p.
method Defining NpL and proving isoperimetric inequality for its dual. result Equivalent to Lp-Busemann-Petty centroid inequality for p=1,2. 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…
We give sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces and use them, for each g at least 2, to identify a constant r_{g-1,2} with the property that the set of closed genus-g hyperbolic surfaces with maximal injectivity radius at least r is compact if and only if r > r_{g-1,2}. T…
Suppose C is a compact, n-edged two-cell of the centered dual decomposition of a locally finite set in the hyperbolic plane, a coarsening of the Delaunay tessellation which was introduced in the author's prior work. We describe an effectively computable lower bound on the area of C, given an n-tuple of positive…
Dual representations for systemic risk measures using acceptance sets.
problem Measuring systemic risk in financial systems.
method Developed dual representations for systemic risk measures based on acceptance sets.
result Simple and self-contained proof of dual representations for utility-based risk measures.
The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk m…
The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily …
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.
New flag area measures constructed via integration over differential forms.
problem Constructing new flag area measures in high-dimensional spaces.
method Using local parallel sets and integration over the normal cycle of differential forms.
result Flag area measures span the space of all smooth SO(n)-covariant flag area measures.
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.
New general volume concept solves Minkowski problem for star bodies.
problem Minkowski problem for star bodies
method General volume concept, new curvature measure, variational formulas, Minkowski-type inequality
result Solution to the Minkowski problem for the new general dual Orlicz curvature measure
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.
Extends dual volume and curvature measures to broader functions and sets, solving Minkowski problems.
problem Characterize measures for which there exists a convex body with a given dual Orlicz curvature measure.
method Extends dual volume and curvature measures to broader functions and sets, proving existence and existence of solutions for Minkowski problems.
result Existence of convex polytopes and solutions for Minkowski problems when measures are discrete or even.
Solves Christoffel problem for disk area measures on spheres.
problem Conditions for a measure to be a disk area measure of convex bodies.
method Integral representation and differential equation reformulation.
result Reconstructs support function from disk area measure.
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.
Classifies nets with area-preserving transformations into two types.
problem Classifying nets with area-preserving transformations.
method Classification using Combescure transformations and isotropic metric duality.
result Found two classes of nets: cone nets and Koenigs nets.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
The Gauss Image Measure uniquely identifies dual convex bodies up to dilation.
problem Identifying dual convex bodies based on their Gauss Image Measure.
method Analyzing the Gauss Image Measure and its properties to establish the uniqueness of dual bodies.
result Dual convex bodies are equal up to a dilation on each path-connected component of the support of the measure.
Investigates set-valued risk measures for processes and vectors, proving equivalence and providing new dual representations.
problem Investigates set-valued risk measures for processes and vectors.
method Utilizes equivalence of risk measures for processes and vectors and their penalty function formulations.
result Provides new dual representation for risk measures for processes in the set-valued framework.
Geometric QCD framework establishes stable vacuum for quark confinement.
problem Quark confinement in QCD.
method Geometric construction of stable vacuum using Hodge-dual surfaces.
result Existence and stability of the Hodge-dual surface in 4D ensures quark confinement.
Researchers prove formulas for flag area measures, extending previous work.
problem Proving additive kinematic formulas for flag area measures.
method Introducing an algebraic framework to compute these formulas explicitly.
result Existence and explicit computation of additive kinematic formulas for flag area measures.
The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the…
We maximize the expected utility of terminal wealth in an incomplete market where there are cone constraints on the investor's portfolio process and the utility function is not assumed to be strictly concave or differentiable. We establish the existence of the optimal solutions to the primal and dual problems and their…
Study dual representations for quasiconvex systemic risk measures.
problem Finding dual representations for quasiconvex systemic risk measures.
method Abstract infinite-dimensional setting, explicit formula for penalty function, nonstandard minimax inequality.
result Explicit formula for the penalty function of quasiconvex compositions.
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.
Solves a complex geometric problem for symmetric convex bodies.
problem Conditions for a measure to be the dual curvature measure of a symmetric convex body.
method Variational approach using entropy and quermassintegrals, with estimates on entropy and curvature measures.
result Explicit conditions for the measure concentration, leading to a full solution for 1<q<n. 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.
In this paper, we obtain the characterizations of Mannheim offsets of the timelike ruled surface with spacelike rulings in dual Lorentzian space. We give the relations between terms of their integral invariants and also we give the new characterization of the Mannheim offsets of developable timelike ruled surface. More…
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.
New convex programs solve minimal-area problems on Riemann surfaces.
problem Finding the conformal metric of least area with constraints on curve lengths.
method Formulated as a local convex program using calibrations and max flow-min cut theorem.
result Derives new insights and numerical solutions for minimal-area metrics.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
New geometric measure simplifies complex analysis.
problem Complex geometric analysis challenges.
method Geometric integration and convergence methods.
result Smallest measure satisfying Area Formula.