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.
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48 results for “Brunn Minkowski theory”
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.
New Steiner formula for affine surface area in Minkowski theory.
problem Developing a new Steiner formula for affine surface area.
method Proving a new Steiner formula for the affine surface area of a Minkowski outer parallel body.
result New curvature measures with properties not previously seen in literature.
Ph.D. thesis on complex Brunn-Minkowski theory using Hilbert bundles.
problem Complex Brunn-Minkowski theory.
method Hilbert bundles approach.
result New insights into complex Brunn-Minkowski theory.
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…
-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 .
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…
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.
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.
This is a survey of results on positivity of vector bundles, inspired by the Brunn-Minkowski and Prékopa theorems. Applications to complex analysis, Kähler geometry and algebraic geometry are also discussed.
Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
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…
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
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.
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
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…
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…
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.
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.
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.
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 .
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.
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.
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.
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.
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 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…
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.
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 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.
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 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.
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.
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 general volume of a star body, a notion that includes the usual volume, the th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the -dual curvature measures introduced recently by Lutwak, Ya…
In this paper we revisit the anisotropic isoperimetric and the Brunn-Minkowski inequalities for convex sets. The best known constant depending on the space dimension in both inequalities is due to Segal [\ref{bib:Seg.}]. We improve that constant to for convex sets and to for centrally sy…
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.
We prove that the log-Brunn-Minkowski inequality \begin{equation*} |λK+_0 (1-λ)L|\geq |K|^λ|L|^{1-λ} \end{equation*} (where is the Lebesgue measure and is the so-called log-addition) holds when is a ball and is a symmetric convex body in a suitable neighborhood of .
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.
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 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.
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…
Paper solves Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
problem Solving Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
method Combining covolume, Hadamard variational formula, and geometric interpretation.
result Solved Minkowski problem for non-compact convex sets under asymptotic conditions.
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 paper solves a geometric problem using curvature flow and variational methods.
problem The -Gaussian Minkowski problem in the Euclidean space.
method Gauss curvature flow and Aleksandrov's variational method with Lagrange multipliers.
result The flow converges to a smooth solution of the -Gaussian Minkowski problem.
Paper solves a generalized chord Minkowski problem using Gauss curvature flows.
problem Generalized chord Minkowski problem using Orlicz functions.
method Gauss curvature flows to achieve existence of smooth solutions.
result Existence of smooth solutions to the orlicz chord Minkowski problem.
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…
We study a Riemannian manifold equipped with a density which satisfies the Bakry--Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is local…