Journal of the American Mathematical Society, ISSN 0894-0347, 07/2013, Volume 26, Issue 3, pp. 831 - 852

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the...

Minkowski problem | Cone-volume measure | Log-Minkowski problem | Lp-Minkowski problem | L-0-MINKOWSKI PROBLEM | log-Minkowski problem | SURFACE MEASURE | P SOBOLEV INEQUALITIES | POLYTOPES | AFFINE ISOPERIMETRIC-INEQUALITIES | MATHEMATICS | L-p-Minkowski problem | CONVEX HYPERSURFACES | CURVATURE | BODIES | EQUATION | GEOMETRY

Minkowski problem | Cone-volume measure | Log-Minkowski problem | Lp-Minkowski problem | L-0-MINKOWSKI PROBLEM | log-Minkowski problem | SURFACE MEASURE | P SOBOLEV INEQUALITIES | POLYTOPES | AFFINE ISOPERIMETRIC-INEQUALITIES | MATHEMATICS | L-p-Minkowski problem | CONVEX HYPERSURFACES | CURVATURE | BODIES | EQUATION | GEOMETRY

Journal Article

中国科学：数学英文版, ISSN 1674-7283, 2017, Volume 60, Issue 10, pp. 1857 - 1872

We prove some analogs inequalities of the logarithmic Minkowski inequality for general nonsymmetric convex bodies...

闵可夫斯基 | 凸体 | 非对称 | 等周不等式 | 对数不等式 | 仿射 | 类似 | p-affine isoperimetric inequality | 52A20 | 52A40 | log-Minkowski inequality | dual log-Minkowski inequality | Mathematics | 52A04 | Applications of Mathematics | Minkowski inequality | L-0-MINKOWSKI PROBLEM | MATHEMATICS | MATHEMATICS, APPLIED | FIREY THEORY | CONE-VOLUME MEASURE | DUAL MIXED VOLUMES | AFFINE | POLYTOPES

闵可夫斯基 | 凸体 | 非对称 | 等周不等式 | 对数不等式 | 仿射 | 类似 | p-affine isoperimetric inequality | 52A20 | 52A40 | log-Minkowski inequality | dual log-Minkowski inequality | Mathematics | 52A04 | Applications of Mathematics | Minkowski inequality | L-0-MINKOWSKI PROBLEM | MATHEMATICS | MATHEMATICS, APPLIED | FIREY THEORY | CONE-VOLUME MEASURE | DUAL MIXED VOLUMES | AFFINE | POLYTOPES

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 05/2012, Volume 262, Issue 9, pp. 4181 - 4204

We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality for entropy...

Logarithmic Sobolev inequality | Affine isoperimetric inequality | MATHEMATICS | Equality

Logarithmic Sobolev inequality | Affine isoperimetric inequality | MATHEMATICS | Equality

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 07/2016, Volume 271, Issue 2, pp. 454 - 473

We show that the Lp Busemann–Petty centroid inequality provides an elementary and powerful tool to the study of some sharp affine functional inequalities with a geometric content, like log-Sobolev, Sobolev and Gagliardo...

Affine Sobolev inequalities | [formula omitted] Busemann–Petty centroid inequality | Affine logarithmic inequalities | Busemann-Petty centroid inequality

Affine Sobolev inequalities | [formula omitted] Busemann–Petty centroid inequality | Affine logarithmic inequalities | Busemann-Petty centroid inequality

Journal Article

Advances in Mathematics, ISSN 0001-8708, 09/2014, Volume 262, pp. 909 - 931

The logarithmic Minkowski problem asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the cone-volume measure of a convex body...

Polytope | [formula omitted] Minkowski problem | Minkowski problem | Cone-volume measure | Monge–Ampère equation | Logarithmic Minkowski problem | Monge-ampère equation | Logarithmic minkowski problem | L-0-MINKOWSKI PROBLEM | MATHEMATICS | REGULARITY | CONVEX HYPERSURFACES | MEAN-CURVATURE | P SOBOLEV INEQUALITIES | BODIES | EQUATION | AFFINE ISOPERIMETRIC-INEQUALITIES | FLOW | GEOMETRY

Polytope | [formula omitted] Minkowski problem | Minkowski problem | Cone-volume measure | Monge–Ampère equation | Logarithmic Minkowski problem | Monge-ampère equation | Logarithmic minkowski problem | L-0-MINKOWSKI PROBLEM | MATHEMATICS | REGULARITY | CONVEX HYPERSURFACES | MEAN-CURVATURE | P SOBOLEV INEQUALITIES | BODIES | EQUATION | AFFINE ISOPERIMETRIC-INEQUALITIES | FLOW | GEOMETRY

Journal Article

JOURNAL OF FUNCTIONAL ANALYSIS, ISSN 0022-1236, 07/2016, Volume 271, Issue 2, pp. 454 - 473

We show that the L-p Busemann-Petty centroid inequality provides an elementary and powerful tool to the study of some sharp affine functional inequalities with a geometric content, like log-Sobolev...

MATHEMATICS | Affine Sobolev inequalities | L-p Busemann-Petty centroid inequality | HYPERCONTRACTIVITY | HAMILTON-JACOBI EQUATIONS | Affine logarithmic inequalities | ISOPERIMETRIC-INEQUALITIES

MATHEMATICS | Affine Sobolev inequalities | L-p Busemann-Petty centroid inequality | HYPERCONTRACTIVITY | HAMILTON-JACOBI EQUATIONS | Affine logarithmic inequalities | ISOPERIMETRIC-INEQUALITIES

Journal Article

Advances in Applied Mathematics, ISSN 0196-8858, 02/2016, Volume 73, pp. 43 - 58

We validate the conjectured logarithmic Minkowski inequality, and thus the equivalent logarithmic Brunn...

Log-Brunn–Minkowski inequality | Log-Minkowski inequality | Cone-volume measure | [formula omitted]-Minkowski problem | Minkowski problem | Log-Brunn-Minkowski inequality | MATHEMATICS, APPLIED | FIREY THEORY | EVOLVING PLANE-CURVES | CURVATURE | AFFINE | L-0-Minkowski problem | Equality

Log-Brunn–Minkowski inequality | Log-Minkowski inequality | Cone-volume measure | [formula omitted]-Minkowski problem | Minkowski problem | Log-Brunn-Minkowski inequality | MATHEMATICS, APPLIED | FIREY THEORY | EVOLVING PLANE-CURVES | CURVATURE | AFFINE | L-0-Minkowski problem | Equality

Journal Article

Geometriae Dedicata, ISSN 0046-5755, 12/2018, Volume 197, Issue 1, pp. 33 - 48

The logarithmic John ellipsoid of a convex body in with its centroid at the origin is introduced by solving a pair of dual optimization problems...

John ellipsoid | Volume-ratio inequality | MATHEMATICS | POSITIONS | EXTREMAL PROBLEMS | SURFACE-AREA | MINKOWSKI-FIREY THEORY | VOLUME | AFFINE | L-p John ellipsoid

John ellipsoid | Volume-ratio inequality | MATHEMATICS | POSITIONS | EXTREMAL PROBLEMS | SURFACE-AREA | MINKOWSKI-FIREY THEORY | VOLUME | AFFINE | L-p John ellipsoid

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 07/2015, Volume 427, Issue 2, pp. 905 - 929

The Orlicz–Brunn–Minkowski theory received considerable attention recently, and many results in the Lp-Brunn–Minkowski theory have been extended to their...

Orlicz–Brunn–Minkowski theory | Geominimal surface area | The Blaschke–Santaló inequality | Affine surface area | Affine isoperimetric inequalities | The inverse Santaló inequality | Orlicz-Brunn-Minkowski theory | The Blaschke-Santaló inequality | MATHEMATICS, APPLIED | The inverse Santalo inequality | SURFACE-AREA | PROOF | INVARIANT VALUATIONS | The Blaschke-Santalo inequality | MATHEMATICS | MINKOWSKI-FIREY THEORY | BODIES

Orlicz–Brunn–Minkowski theory | Geominimal surface area | The Blaschke–Santaló inequality | Affine surface area | Affine isoperimetric inequalities | The inverse Santaló inequality | Orlicz-Brunn-Minkowski theory | The Blaschke-Santaló inequality | MATHEMATICS, APPLIED | The inverse Santalo inequality | SURFACE-AREA | PROOF | INVARIANT VALUATIONS | The Blaschke-Santalo inequality | MATHEMATICS | MINKOWSKI-FIREY THEORY | BODIES

Journal Article

Journal of the Mathematical Society of Japan, ISSN 0025-5645, 2017, Volume 69, Issue 4, pp. 1565 - 1581

We reduce Iitaka's subadditivity conjecture for the logarithmic Kodaira dimension to a special case of the generalized abundance conjecture by establishing an Iitaka type inequality for Nakayama's...

Affine varieties | ω-sheaf | Abundance conjecture | Iitaka conjecture | Minimal model program | Nakayama's numerical Kodaira dimension | Logarithmic Kodaira dimension | MATHEMATICS | logarithmic Kodaira dimension | abundance conjecture | affine varieties | (omega)over-cap-sheaf | MINIMAL MODELS | omega-sheaf | minimal model program

Affine varieties | ω-sheaf | Abundance conjecture | Iitaka conjecture | Minimal model program | Nakayama's numerical Kodaira dimension | Logarithmic Kodaira dimension | MATHEMATICS | logarithmic Kodaira dimension | abundance conjecture | affine varieties | (omega)over-cap-sheaf | MINIMAL MODELS | omega-sheaf | minimal model program

Journal Article

Advances in Mathematics, ISSN 0001-8708, 10/2016, Volume 302, pp. 1080 - 1110

.... Our approach is based on the general Lp Busemann–Petty centroid inequality and does not rely on the general Lp Petty projection inequality or the solution of the Lp Minkowski problem. A Brothers...

Busemann–Petty centroid inequality | Affine Sobolev-type inequalities | Affine Pólya–Szegö principle | Stability estimates | REARRANGEMENT | MATHEMATICS | Busemann-Petty centroid inequality | SHARP SOBOLEV | CONVEX SYMMETRIZATION | Affine Polya-Szego principle | Equality

Busemann–Petty centroid inequality | Affine Sobolev-type inequalities | Affine Pólya–Szegö principle | Stability estimates | REARRANGEMENT | MATHEMATICS | Busemann-Petty centroid inequality | SHARP SOBOLEV | CONVEX SYMMETRIZATION | Affine Polya-Szego principle | Equality

Journal Article

Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 05/2020, Volume 545, p. 123270

.... We comment on the extremizing probability distributions of this LYZ functional, its relation to the escort distributions, a generalized Fisher information and the corresponding Cramér–Rao inequality. We point out potential physical implications of the LYZ entropic functional and of its extremal distributions.

Nonextensive thermostatistics | Nonadditive entropy | Tsallis entropy | [formula omitted]-entropy | Complexity | FOKKER-PLANCK EQUATION | PHYSICS, MULTIDISCIPLINARY | INEQUALITY | SOBOLEV | MINKOWSKI-FIREY THEORY | IRREVERSIBILITY | GENERALIZED ENTROPIES | q-entropy | AFFINE | H-THEOREM | CRAMER-RAO

Nonextensive thermostatistics | Nonadditive entropy | Tsallis entropy | [formula omitted]-entropy | Complexity | FOKKER-PLANCK EQUATION | PHYSICS, MULTIDISCIPLINARY | INEQUALITY | SOBOLEV | MINKOWSKI-FIREY THEORY | IRREVERSIBILITY | GENERALIZED ENTROPIES | q-entropy | AFFINE | H-THEOREM | CRAMER-RAO

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2018, Volume 2018, Issue 1, pp. 1 - 9

...}$-mixed volume and the Lp $L_{p}$-dual mixed volume. In this article, associated with the (p,q) $(p,q)$-mixed volumes, we establish related cyclic inequalities, monotonic inequalities, and product inequalities.

52A20 | Cyclic inequality | 52A40 | ( p , q ) $(p,q)$ -mixed volume | Analysis | Monotonic inequality | 52A39 | Mathematics, general | Mathematics | Applications of Mathematics | Product inequality | (p, q) -mixed volume | MATHEMATICS | MATHEMATICS, APPLIED | AFFINE SURFACE-AREA | (p, q)-mixed volume | MINKOWSKI-FIREY THEORY | MIXED VOLUMES | Inequalities | Cases (containers)

52A20 | Cyclic inequality | 52A40 | ( p , q ) $(p,q)$ -mixed volume | Analysis | Monotonic inequality | 52A39 | Mathematics, general | Mathematics | Applications of Mathematics | Product inequality | (p, q) -mixed volume | MATHEMATICS | MATHEMATICS, APPLIED | AFFINE SURFACE-AREA | (p, q)-mixed volume | MINKOWSKI-FIREY THEORY | MIXED VOLUMES | Inequalities | Cases (containers)

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 2019, Volume 2019, Issue 1, pp. 1 - 21

Böröczky et al. proposed the log-Minkowski problem and established the plane log-Minkowski inequality for origin-symmetric convex bodies...

mixed quermassintegral | Dual log-Minkowski inequality | Log-Minkowski inequality | dual mixed quermassintegral | mixed affine isoperimetric inequality | Differential geometry | Inequalities | Inequality

mixed quermassintegral | Dual log-Minkowski inequality | Log-Minkowski inequality | dual mixed quermassintegral | mixed affine isoperimetric inequality | Differential geometry | Inequalities | Inequality

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 02/2019, Volume 371, Issue 2, pp. 971 - 1002

.... When the logarithmic Kodaira dimension \bar {\kappa } is -\infty , we completely characterize the potential density of integral points in terms of the number of irreducible...

MATHEMATICS | COMPLEMENT | SUBVARIETIES | DIOPHANTINE APPROXIMATION | CURVE | VARIETIES | AFFINE | LOGARITHMIC KODAIRA DIMENSION

MATHEMATICS | COMPLEMENT | SUBVARIETIES | DIOPHANTINE APPROXIMATION | CURVE | VARIETIES | AFFINE | LOGARITHMIC KODAIRA DIMENSION

Journal Article

Geometriae Dedicata, ISSN 0046-5755, 8/2015, Volume 177, Issue 1, pp. 75 - 82

Recently Böröczky, Lutwak, Yang and Zhang have proved the log-Brunn–Minkowski inequality which is stronger than the classical Brunn...

Geometry | L_{0}$$ L 0 -Minkowski addition | Log-Brunn–Minkowski inequality | 52A40 | Log-Minkowski inequality | Minkowski mixed-volume inequality | Mathematics | (Formula presented.)-Minkowski addition | MATHEMATICS | FIREY THEORY | L-0-Minkowski addition | SURFACE MEASURE | SPHERE | CURVATURE | CLASSIFICATION | AFFINE | Log-Brunn-Minkowski inequality | Equality

Geometry | L_{0}$$ L 0 -Minkowski addition | Log-Brunn–Minkowski inequality | 52A40 | Log-Minkowski inequality | Minkowski mixed-volume inequality | Mathematics | (Formula presented.)-Minkowski addition | MATHEMATICS | FIREY THEORY | L-0-Minkowski addition | SURFACE MEASURE | SPHERE | CURVATURE | CLASSIFICATION | AFFINE | Log-Brunn-Minkowski inequality | Equality

Journal Article

Discrete & Computational Geometry, ISSN 0179-5376, 9/2012, Volume 48, Issue 2, pp. 281 - 297

Quite recently, an Orlicz Minkowski problem has been posed and the existence part of this problem for even measures has been presented. In this paper, the...

Convex polytope | Computational Mathematics and Numerical Analysis | Mathematics | Minkowski problem | Combinatorics | Orlicz norm | L-0-MINKOWSKI PROBLEM | VOLUME INEQUALITIES | FIREY THEORY | VALUED VALUATIONS | SURFACE-AREA | ELLIPSOIDS | P SOBOLEV INEQUALITIES | AFFINE ISOPERIMETRIC-INEQUALITIES | MATHEMATICS | PETTY CENTROID INEQUALITY | ISOTROPIC CONVEX-BODIES | COMPUTER SCIENCE, THEORY & METHODS | Polytopes | Geometry | Theorems | Computational geometry

Convex polytope | Computational Mathematics and Numerical Analysis | Mathematics | Minkowski problem | Combinatorics | Orlicz norm | L-0-MINKOWSKI PROBLEM | VOLUME INEQUALITIES | FIREY THEORY | VALUED VALUATIONS | SURFACE-AREA | ELLIPSOIDS | P SOBOLEV INEQUALITIES | AFFINE ISOPERIMETRIC-INEQUALITIES | MATHEMATICS | PETTY CENTROID INEQUALITY | ISOTROPIC CONVEX-BODIES | COMPUTER SCIENCE, THEORY & METHODS | Polytopes | Geometry | Theorems | Computational geometry

Journal Article

Journal of Inequalities and Applications, 12/2019, Volume 2019, Issue 1, pp. 1 - 21

Böröczky et al. proposed the log-Minkowski problem and established the plane log-Minkowski inequality for origin-symmetric convex bodies...

52A20 | L p $L_{p}$ -dual mixed quermassintegral | L p $L_{p}$ -mixed affine isoperimetric inequality | 52A40 | Dual log-Minkowski inequality | Log-Minkowski inequality | Analysis | Mathematics, general | Mathematics | L p $L_{p}$ -mixed quermassintegral | Applications of Mathematics

52A20 | L p $L_{p}$ -dual mixed quermassintegral | L p $L_{p}$ -mixed affine isoperimetric inequality | 52A40 | Dual log-Minkowski inequality | Log-Minkowski inequality | Analysis | Mathematics, general | Mathematics | L p $L_{p}$ -mixed quermassintegral | Applications of Mathematics

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2007, Volume 17, Issue 1, pp. 119 - 146

...SIAM J. OPTIM. c Vol. 17, No. 1, pp. 119146 CONSTRAINT REDUCTION FOR LINEAR PROGRAMS WITH MANY INEQUALITY CONSTRAINTS ANDR L. TITS, P.-A. ABSIL, AND WILLIAM P...

Linear programming | Column generation | Constraint reduction | Mehrotra's predictor-corrector | Affine scaling | Primal-dual interior-point methods | MATHEMATICS, APPLIED | COMPLEXITY ANALYSIS | primal-dual interior-point methods | ALGORITHM | column generation | linear programming | affine scaling | constraint reduction

Linear programming | Column generation | Constraint reduction | Mehrotra's predictor-corrector | Affine scaling | Primal-dual interior-point methods | MATHEMATICS, APPLIED | COMPLEXITY ANALYSIS | primal-dual interior-point methods | ALGORITHM | column generation | linear programming | affine scaling | constraint reduction

Journal Article

Management Science, ISSN 0025-1909, 12/1995, Volume 41, Issue 12, pp. 1922 - 1934

.... In particular none of them offer a simple, self-contained introduction to the theory of Linear Programming and linear inequalities...

linear inequalities | complementarity | Farkas lemma | unboundedness | duality | boundedness | interior points | Linear Programming | Management principles | Interior points | Optimal solutions | Mathematical theorems | Algorithms | Simplex method | Linear programming | Mathematical duality | Mathematical vectors | Index sets | MANAGEMENT | TRAJECTORIES | ALGORITHMS | Farkas Lemma | LIMITING BEHAVIOR | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | NONLINEAR GEOMETRY | AFFINE | linear inequalities; Linear Programming; interior points; duality; complementarity; Farkas lemma; boundedness; unboundedness | Analysis | Duality theory (Mathematics) | Studies | Management science | Mathematical analysis

linear inequalities | complementarity | Farkas lemma | unboundedness | duality | boundedness | interior points | Linear Programming | Management principles | Interior points | Optimal solutions | Mathematical theorems | Algorithms | Simplex method | Linear programming | Mathematical duality | Mathematical vectors | Index sets | MANAGEMENT | TRAJECTORIES | ALGORITHMS | Farkas Lemma | LIMITING BEHAVIOR | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | NONLINEAR GEOMETRY | AFFINE | linear inequalities; Linear Programming; interior points; duality; complementarity; Farkas lemma; boundedness; unboundedness | Analysis | Duality theory (Mathematics) | Studies | Management science | Mathematical analysis

Journal Article

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