Paper proves log-Brunn-Minkowski inequality in 3D space.
problem Proving stronger geometric inequalities in higher dimensions.
method Establishes inequalities for convex bodies in R^3.
result Log-Brunn-Minkowski inequality holds in 3D.
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48 results for “Brunn–Minkowski inequalities”
Proves equivalence of local and global -Brunn-Minkowski inequalities.
problem Equivalence of local and global -Brunn-Minkowski inequalities.
method Study of -combinations of strongly isomorphic polytopes.
result Equivalence of -Brunn-Minkowski inequalities proved.
Magnetic Brunn-Minkowski inequalities on Riemannian manifolds
problem Establishing equivalence between Brunn-Minkowski inequalities and magnetic Ricci curvature
method Using magnetic geodesics
result Proving a sharp, undistorted Brunn-Minkowski inequality
-Brunn-Minkowski inequality for math.AP
Proves global -Brunn-Minkowski inequality for specific range.
problem Establishing global -Brunn-Minkowski inequality for in a specific range.
method Using local uniqueness results for the -Minkowski problem and Logarithmic Minkowski problem.
result Global -Brunn-Minkowski inequality proven for .
SubRiemannian structures fail to meet Riemannian Brunn--Minkowski inequalities.
problem SubRiemannian structures do not satisfy Riemannian Brunn--Minkowski inequalities.
method The proof relies on the method used for the Heisenberg group and new investigations by Agrachev, Barillari, and Rizzi on subRiemannian structures.
result No Brunn--Minkowski inequality can be satisfied by strictly subRiemannian structures.
Log-Brunn-Minkowski inequality proven for ball and symmetric convex bodies.
problem Proving the log-Brunn-Minkowski inequality for specific geometric shapes.
method Analyzing the inequality for a ball and symmetric convex bodies in a neighborhood.
result Log-Brunn-Minkowski inequality holds for a ball and symmetric convex bodies.
This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz-Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is…
Study confirms equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.
problem Equivalence between curvature-dimension conditions and strong Brunn-Minkowski inequalities in Heisenberg groups.
method Optimal transport and approximation techniques in sub-Riemannian Heisenberg group Hn, combined with previous works.
result Confirms the equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.
Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of resu…
The study generalizes Minkowski inequalities for curves on surfaces.
problem Generalizing Minkowski inequalities for curves on Riemannian surfaces.
method Introducing a generalized Minkowski average using parametrized curves and proving the inequality for constant-speed curves.
result A family of constant-speed curves on a Riemannian surface satisfies the Brunn-Minkowski inequality with respect to the Riemannian area form if and only if the geodesic curvature is determined by a function κ satisfying a specific inequality.
Local logarithmic Brunn-Minkowski holds for zonoids.
problem Logarithmic Brunn-Minkowski conjecture for zonoids
method Bochner method variant
result Local form of conjecture proven for zonoids
New proof of log-Brunn-Minkowski inequality for zonoids and convex bodies.
problem Proving the log-Brunn-Minkowski inequality for convex bodies and zonoids.
method Establishing monotonicity of the deficit in the LLBM under line segment addition.
result Equality in LLBM for smooth convex bodies occurs only for homothetic bodies.
Timelike curvature and Brunn-Minkowski inequality linked in non-smooth spacetimes.
problem Equivalence between timelike Ricci curvature and Brunn-Minkowski inequality in synthetic Lorentzian spaces.
method Introducing strong -timelike Brunn-Minkowski condition and proving equivalence to curvature conditions.
result Timelike curvature dimension condition equivalent to timelike Brunn-Minkowski inequality in specific settings.
Unified study of Brunn-Minkowski conjectures for log-concave measures.
problem Understanding the role of symmetry in inequalities of Brunn-Minkowski type.
method Unified framework, new results for conjectures, improved estimates for Lebesgue and Gaussian measures.
result Unified framework and new results for Brunn-Minkowski conjectures.
We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.
The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.
problem Rigidity of inequalities on weighted Riemannian manifolds.
method Theorems of rigidity on curvature and measure for the Borell-Brascamp-Lieb inequality, generalizing a theorem by Balogh and Kristály.
result A generalization of the curvature rigidity theorem to the weighted setting.
Proves inequality for special 3D shapes, generalizing to non-symmetric ones.
problem Proving a mathematical inequality for specific 3D shapes.
method Operator theoretic approach combined with spherical function decomposition.
result Generalized inequality for non-symmetric bodies of revolution.
Paper proves Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
problem Proving equivalence between Brunn-Minkowski inequality and curvature dimension condition.
method Analyzes weighted Riemannian manifolds, proving equivalence without optimal transport or differential structure.
result Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz -radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz -radial addition of two star bodies, we derive a f…
The paper examines the failure of Brunn-Minkowski inequality for certain convex bodies.
problem Brunn-Minkowski inequality for -th dual quermassintegrals with .
method Second variation argument, dimension reduction, Hadwiger's inequality, singular weighted Reilly formula, coordinate-slice Hardy inequality.
result Established the inequality for unconditional convex bodies in the full range .
Paper proves uniqueness of solutions to a geometric inequality problem.
problem Uniqueness of solutions to the isotropic Minkowski problem.
method Analysis of the Hilbert-Brunn-Minkowski operator to derive stability estimates.
result Uniqueness of -isotropic solutions to the isotropic Minkowski problem in for specific ranges of .
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.
New geometric inequalities for convex bodies derived from Log-Brunn-Minkowski conjecture.
problem Proving geometric inequalities for convex bodies.
method Analyzing semi-norms and symmetric convex bodies, using integral inequalities.
result Characterization and improvement of geometric inequalities involving convex bodies.
The present note is a result of an on-going investigation into the logarithmic Brunn-Minkowski inequality. We obtain lower estimates on the volume product for convex bodies in not necessarily symmetric with respect to the origin from a modified logarithmic Brunn-Minkowski inequality.
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.
This paper aims to develop basic theory for the dual Orlicz 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…
Paper improves constants in anisotropic inequalities for convex sets.
problem Improving constants in anisotropic isoperimetric and Brunn-Minkowski inequalities.
method Combining mass transportation, geometric-arithmetic mean inequalities, and trace estimates.
result Best constants conjectured to be quadratic in n.
Study finds non-uniqueness for Minkowski problem solutions in higher dimensions, uniqueness in lower dimensions.
problem Minkowski problem of even dual curvature measures in higher dimensions.
method Logarithmic Brunn-Minkowski inequality for dual quermassintegrals.
result Non-uniqueness for and uniqueness for in .
The paper proves inequalities for hyperbolic sets and curves.
problem Proving inequalities for sets and curves in hyperbolic geometry.
method Defining horocyclic Minkowski sums and proving inequalities for hyperbolic areas.
result Horocyclic Brunn-Minkowski inequality holds for hyperbolic sets.
New inequalities on Riemannian manifolds with weighted densities.
problem Inequalities on Riemannian manifolds with weighted densities.
method Poincaré and Brunn–Minkowski inequalities on weighted Riemannian manifolds.
result Novel Brunn–Minkowski inequality in the weighted-Riemannian setting.
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.
The hyperbolic space $ \H^d$ can be defined as a pseudo-sphere in the Minkowski space-time. In this paper, a Fuchsian group is a group of linear isometries of the Minkowski space such that $\H^d/Γ$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, g…
New proof shows origin-centred balls are unique solutions to curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Local Brunn-Minkowski inequality and Alexandrov-Fenchel inequality.
result Origin-centred balls are the only solutions to curvature and related problems.
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
New insights from centro-affine geometry solve a key geometric conjecture.
problem Log-Brunn-Minkowski conjecture in centro-affine differential geometry.
method Interpreting the log-Brunn-Minkowski conjecture as a spectral problem and using centro-affine differential geometry.
result Global uniqueness and inequalities in the log-Minkowski problem for certain convex bodies.
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 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.
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.
The paper proves a Gaussian measure version of the Brunn-Minkowski inequality.
problem Proving a Brunn-Minkowski inequality for Gaussian measure.
method Raywise radial-tangential localization of the Hessian energy of a solution to a Neumann problem.
result The largest number α_γ(n) for the inequality is found to be 1 - 2/(n-1) * (Γ(n/2)^2 / Γ((n-1)/2)^2).
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…
Paper solves capillary Orlicz-Minkowski problem with new inequalities.
problem Finding capillary convex bodies with prescribed Orlicz surface area measures.
method Continuity method and inequalities to solve the capillary even Orlicz-Minkowski problem.
result Volume-normalized smooth solutions and inequalities established.
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…
A sharp centro-affine isospectral inequality of Szegö--Weinberger type and the -Minkowski problemmath.FA
Sharp inequality for eigenvalues of convex bodies, proving ellipsoid uniqueness.
problem Eigenvalue bounds for convex bodies and ellipsoid uniqueness.
method Established a sharp upper-bound for eigenvalues of a specific operator.
result Equality holds only for ellipsoids, complementing conjectural lower-bound.
Survey on positivity in vector bundles, inspired by Brunn-Minkowski theorems.
problem Positivity of vector bundles in complex analysis, Kähler geometry, and algebraic geometry.
method Inspiration from Brunn-Minkowski theorems.
result Survey results on positivity in vector bundles.
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 .
The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the -Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz affine and geominimal surface areas for single convex body as well as for multiple convex bod…
We introduce the dual isoperimetrix which solves the isoperimetric problem in the dual Brunn-Minkowski theory. We then show how the dual isoperimetrix is related to the isoperimetrix from the Brunn-Minkowski theory.
The paper proves a Gaussian measure's concavity for symmetric convex sets up to a factor of 2.
problem Proving the concavity of the Gaussian measure for symmetric convex sets.
method Analyzing the conjecture of Gardner and Zvavitch, proving the inequality up to a factor of 2.
result The Gaussian measure satisfies a factor of 2 concavity inequality for symmetric convex sets.