A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations.
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48 results for “Minkowski valuations”
A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwi…
New Orlicz Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are…
Fixed points of Minkowski valuations are found in specific ball neighborhoods.
problem Finding fixed points of Minkowski valuations.
method Lutwak-Schneider class reduction technique and Petty's conjectured projection inequality.
result Balls are the only solutions to the fixed-point problem for certain Minkowski valuations.
A complete classification of all continuous GL(n) contravariant Minkowski valuations is established. As an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical Petty projection inequality.
The space of Minkowski valuations on an m-dimensional complex vector space which are continuous, translation invariant and contravariant under the complex special linear group is explicitly described. Each valuation with these properties is shown to satisfy geometric inequalities of Brunn-Minkowski, Aleksandrov-Fenchel…
The classification of continuous, translation invariant Minkowski valuations which are contravariant (or covariant) with respect to the complex special linear group is established in a 2-dimensional complex vector space. Every such valuation is given by the sum of a valuation of degree of homogeneity 1 and 3. In dimens…
The decomposition of the space of continuous and translation invariant valuations into a sum of SO(n) irreducible subspaces is obtained. A reformulation of this result in terms of a Hadwiger type theorem for continuous translation invariant and SO(n)-equivariant tensor valuations is also given. As an application, symme…
The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only non-trivi…
A complete classification is obtained of continuous, translation invariant, Minkowski valuations on an m-dimensional complex vector space which are covariant under the complex special linear group.
In this paper, we endow the space of continuous translation invariant valuation on convex sets generated by mixed volumes coupled with a suitable Radon measure on tuples of convex bodies with two appropriate norms. This enables us to construct a continuous extension of the convolution operator on smooth valuations to n…
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.
We show that the natural "convolution" on the space of smooth, even, translation-invariant convex valuations on a euclidean space , obtained by intertwining the product and the duality transform of S. Alesker, may be expressed in terms of Minkowski sum. Furthermore the resulting product extends naturally to odd valu…
Fixed points of mean section operators found in convex bodies.
problem Characterizing fixed points of mean section operators in convex bodies.
method Characterization of rotation equivariant operators using spherical Laplacian mass distribution, and application of Minkowski valuations.
result Euclidean balls are the only fixed points of mean section operators in a neighborhood of the unit ball.
Solves Christoffel-Minkowski problem for axially symmetric bodies.
problem Necessary and sufficient conditions for mixed area measures of axially symmetric convex bodies.
method Introduced a new method to transform mixed area measures and mixed volumes of axially symmetric bodies, refining Firey's classification and improving estimates.
result Complete solution to the mixed Christoffel-Minkowski problem for axially symmetric bodies without regularity assumptions.
Study on affine surface areas and their inequalities for convex bodies.
problem Understanding weighted affine surface areas in convex bodies.
method Investigating valuations, isoperimetric inequalities, and connections to divergences.
result Established isoperimetric inequalities for weighted affine surface areas.
Actuarial-consistency and two-step actuarial valuations: a new paradigm to insurance valuationq-fin.PR
Paper introduces new actuarial-consistent valuations for insurance liabilities.
problem Valuation of insurance liabilities considering both financial and actuarial risks.
method Proposes two-step actuarial valuations and actuarial-consistent procedures.
result Actuarial-consistent valuations are equivalent to two-step actuarial valuations under coherence.
Valuation Reveals Uncertaintyq-fin.MF
Paper recovers uncertainty from dynamic valuation rules.
problem Recovering latent uncertainty from observable valuation rules.
method Developed procedures to identify and characterize uncertainty structures from valuation rules.
result Valuation rules contain sufficient information to identify and recover uncertainty structures.
Study convolution of invariant valuations on Lie groups.
problem Understanding convolution of valuations on Lie groups.
method Explicit formula for left-invariant valuations, showing existence of smooth bi-invariant valuations, defining convolution on arbitrary Lie groups.
result Unified convolution operations on Lie groups.
SL(n) covariant valuations on Orlicz spaces are represented and characterized.
problem Representing SL(n) covariant valuations on Orlicz spaces.
method Representation theorem established for continuous, SL(n) covariant vector-valued valuations.
result Unique characterization of SL(n) covariant valuations as moment vectors.
Market valuation duration is 175 years, but drops to 46 years during crises.
problem Understanding the duration of market valuation and its impact on returns.
method Comparing market valuation ratios and dividends to estimate duration, analyzing the discount rate effect.
result Valuation duration is negatively correlated with market returns, with a robust out-of-sample R2 of 15%.
Paper simplifies default process modeling and credit valuation.
problem Modeling and pricing derivative securities with credit risk.
method Integrates default process, probability, and correlation into a unified framework.
result Risky valuation is Martingale in the proposed model.
Business-cycles and Cash-on-Market: Pre-money Startup Valuation in the Macroeconomic Environmentq-fin.GN
Business cycles affect startup valuations, both directly and indirectly.
problem How do business cycles impact startup valuations?
method Structural Equation Model approach using a dataset of 1,089 venture capital investments.
result Business cycles impact startup valuations both directly and indirectly.
Böröczky, Lutwak, Yang and Zhang recently proved the log-Brunn-Minkowski inequality which is stronger than the classical Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper establishes the log-Brunn-Minkowski, log-Minkowski, -Minkowski and -Brunn-Minkowski inequalities f…
Classification of SL(n) covariant valuations on Orlicz spaces.
problem Classifying continuous SL(n) covariant valuations on Orlicz spaces.
method Complete classification without symmetric assumptions, focusing on moment matrix and a new functional in dimension two.
result The moment matrix is the only SL(n) covariant valuation for n≥3, and a new functional appears in dimension two.
We prove new kinematic formulas for tensor valuations and simplify previously known Crofton formulas by using the recently developed algebraic theory of translation invariant valuations. The heart of the paper is the computation of the Alesker-Fourier transform on the large class of spherical valuations, which is achie…
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
Classifies contravariant matrix-valued valuations on polytopes without continuity assumptions.
problem Classifying contravariant matrix-valued valuations on polytopes without continuity assumptions.
method Complete classification of contravariant matrix-valued valuations on polytopes in without continuity assumptions.
result The only such valuation is the general Lutwak-Yang-Zhang matrix in dimension , and a new function in dimension 3.
Let denote the identity connected component of the real orthogonal group with signature . We give a complete description of the spaces of continuous and generalized translation- and -invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invari…
Study evaluates valuation models for UK companies using case studies.
problem Determining how accounting numbers affect business value.
method Comprehensive review of three valuation models: FCFVM, REVM, AEGM.
result Accounting numbers through valuation models can affect business value.
This paper proposes a paradigm shift in the valuation of long term annuities, away from classical no-arbitrage valuation towards valuation under the real world probability measure. Furthermore, we apply this valuation method to two examples of annuity products, one having annual payments linked to a mortality index and…
This paper provides intuition on the relationship of accrual and mark-to-market valuation for cash and forward interest rate trades. Discounted cashflow valuation is compared to spread-based valuation for forward trades, which explains the trader's view on valuation. This is followed by Taylor series approximation for …
Orlicz-Minkowski flowsmath.DG
Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.
problem Anisotropic non-homogeneous Gauss curvature flows and Orlicz-Minkowski problems.
method Long-time existence and behavior analysis, parabolic approximation method, curvature flow.
result Existence and new results for Orlicz-Minkowski problems, including versions.
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 .
Computes tube formulas for valuations in complex space forms.
problem Computing values of valuations on complex space forms.
method Develops tube formulas for valuations in complex space forms and generalizes classical formulas.
result Generalizes classical formulas of Weyl, Gray and others.
Developed a sub-Riemannian version of Minkowski problem in Heisenberg groups.
problem Minkowski type problem in Heisenberg groups.
method Variational method.
result Positive answer to sub-Riemannian Minkowski type problem.
This paper addresses credit valuation adjustment with a new closeout convention.
problem Accurate estimation of financial claim value considering counterparty credit risk.
method Theoretical and computational analysis of a nonlinear valuation system using neural networks.
result A neural network-based algorithm effectively solves the high-dimensional nonlinear valuation system.
Classifies SL(n) covariant matrix-valued valuations on Lp-spaces.
problem Classifying SL(n) covariant matrix-valued valuations on Lp-spaces.
method Established a complete classification for continuous and SL(n) covariant matrix-valued valuations on Lp(Rn,|x|2dx), eliminating matrix symmetry assumption.
result Unique characterization of such valuations by the moment matrix in n>2, rotation matrix in 2D.
Existence of smooth valuations on subspaces is shown for certain conditions.
problem Existence of smooth valuations on subspaces with given restrictions.
method Analyzing compatibility and using recursive descriptions of the cosine transform.
result Compatibility is sufficient for extensibility in certain regimes.
Develops a new method to study algebraic tangent cones of sheaves using valuations.
problem Analyzing tangent cones of torsion-free sheaves on algebraic varieties.
method Introduces a slope stability theory and uses it to define a canonical tangent cone for quasi-regular valuations.
result Shows the existence of a canonical tangent cone for torsion-free sheaves, up to equivalence.
We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral geometry. Then we formulate a conjectural analogue of this result for odd valuations.
Fair market valuations ignore future worker profits in employee-owned firms.
problem Ignoring future worker profits in fair market valuations for employee-owned firms.
method Analyzing property rights and residual claimants in employee-owned firms.
result Fair market valuations are inappropriate for employee-owned firms.
We introduce the new notion of convolution of a (smooth or generalized) valuation on a group and a valuation on a manifold acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and generalized valuations on are modules over the algebra of compactly supported g…
Paper solves Minkowski problem for anisotropic p-torsional rigidity.
problem Solving the Minkowski problem for anisotropic p-torsional rigidity.
method Using the anisotropic -Laplacian equation, presenting sufficient and necessary conditions for existence.
result Presented sufficient and necessary conditions for the existence of a solution.
Develops Brunn-Minkowski theory in hyperbolic space using hyperbolic p-sum.
problem No universally accepted sum for sets in hyperbolic space.
method Introduces hyperbolic p-sum and develops horospherical p-Brunn-Minkowski theory.
result Solves p-Minkowski and p-Christoffel-Minkowski problems for various p.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Enhances data valuation by integrating global and local statistical properties.
problem Insufficient consideration of global and local statistical properties in data valuation methods.
method Proposes a method that fuses global and local statistical properties into regularization terms for Shapley value estimation and dynamic data valuation.
result Demonstrates improved performance and efficiency of data valuation methods through integration of global and local statistical properties.