2015, Mathematical surveys and monographs, ISBN 9781470421939, Volume 202, v.

Book

Statistics & probability letters, ISSN 0167-7152, 2019, Volume 145, pp. 110 - 117

It is known that the distribution of an integrable random vector ξ in Rd is uniquely determined by a (d+1)-dimensional convex body called the lift zonoid of ξ....

Outlier | Lift zonoid | Support function | Risk measure | Selection expectation | Random set | CONVEX HULLS | STATISTICS & PROBABILITY | ZONOIDS

Outlier | Lift zonoid | Support function | Risk measure | Selection expectation | Random set | CONVEX HULLS | STATISTICS & PROBABILITY | ZONOIDS

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2016, Volume 2016, Issue 1

Journal Article

International mathematics research notices, ISSN 1687-0247, 2012, Volume 2012, Issue 1, pp. 1 - 16

Let K be a convex body in R-n with Santalo point at 0. We show that if K has a point on the boundary with positive generalized Gau ss curvature, then the...

MATHEMATICS | BODIES | MINIMAL VOLUME-PRODUCT | ZONOIDS

MATHEMATICS | BODIES | MINIMAL VOLUME-PRODUCT | ZONOIDS

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 06/2015, Volume 426, Issue 1, pp. 1 - 42

Weakly stationary random processes of k-dimensional affine subspaces (flats) in Rn are considered. If 2k≥n, then intersection processes are investigated, while...

Intersection process | Proximity | Associated zonoid | k-flat process | Poisson process | Stability estimate | K-flat process | MATHEMATICS, APPLIED | CONVEX-BODIES | POISSON | STATISTICS | STABILITY | DISTANCES | ZONOIDS | CENTRAL LIMIT-THEOREMS | DISTRIBUTIONS | MATHEMATICS | GEOMETRY

Intersection process | Proximity | Associated zonoid | k-flat process | Poisson process | Stability estimate | K-flat process | MATHEMATICS, APPLIED | CONVEX-BODIES | POISSON | STATISTICS | STABILITY | DISTANCES | ZONOIDS | CENTRAL LIMIT-THEOREMS | DISTRIBUTIONS | MATHEMATICS | GEOMETRY

Journal Article

The Journal of geometric analysis, ISSN 1559-002X, 2018, Volume 29, Issue 3, pp. 2998 - 3009

The cosine representation of the support function of a centrally symmetric convex body plays a fundamental role in integral geometry. In this article, one new...

52A20 | 53C45 | 53C65 | Mathematics | Convex body | Abstract Harmonic Analysis | Integral geometry | Fourier Analysis | Zonoid | Convex and Discrete Geometry | Global Analysis and Analysis on Manifolds | Support function | Differential Geometry | Dynamical Systems and Ergodic Theory | MATHEMATICS | ZONOIDS

52A20 | 53C45 | 53C65 | Mathematics | Convex body | Abstract Harmonic Analysis | Integral geometry | Fourier Analysis | Zonoid | Convex and Discrete Geometry | Global Analysis and Analysis on Manifolds | Support function | Differential Geometry | Dynamical Systems and Ergodic Theory | MATHEMATICS | ZONOIDS

Journal Article

Journal of Multivariate Analysis, ISSN 0047-259X, 01/2016, Volume 143, pp. 394 - 397

Under some mild conditions on probability distribution P, if limnPn=P weakly then the sequence of zonoid depth functions with respect to Pn converges uniformly...

Zonoid depth | Uniform consistency | STATISTICS & PROBABILITY

Zonoid depth | Uniform consistency | STATISTICS & PROBABILITY

Journal Article

Duke Mathematical Journal, ISSN 0012-7094, 2014, Volume 163, Issue 11, pp. 2003 - 2022

In this note we link symplectic and convex geometry by relating two seemingly_different open conjectures: a symplectic isoperimetric-type inequality for convex...

TOPOLOGY | MATHEMATICS | MINIMAL VOLUME-PRODUCT | SPACES | ZONOIDS | GEOMETRY | 52A20 | 52A40 | 37D50 | 52A23

TOPOLOGY | MATHEMATICS | MINIMAL VOLUME-PRODUCT | SPACES | ZONOIDS | GEOMETRY | 52A20 | 52A40 | 37D50 | 52A23

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 12/2016, Volume 368, Issue 12, pp. 8873 - 8899

Bull. AMS (2001)]. We show that the k-th intrinsic volume of the set of all functions on [0,1] which have Lipschitz constant bounded by 1 and which vanish at 0...

Sobolev balls | Sudakov’s formula | Intrinsic volumes | Mean width | Tsirelson’s theorem | Gaussian processes | Brownian zonoids | Brownian convex hulls | Lipschitz balls | Ellipsoids | MATHEMATICS | Sudakov's formula | mean width | Tsirelson's theorem | ellipsoids

Sobolev balls | Sudakov’s formula | Intrinsic volumes | Mean width | Tsirelson’s theorem | Gaussian processes | Brownian zonoids | Brownian convex hulls | Lipschitz balls | Ellipsoids | MATHEMATICS | Sudakov's formula | mean width | Tsirelson's theorem | ellipsoids

Journal Article

Geometric and functional analysis, ISSN 1420-8970, 2008, Volume 18, Issue 3, pp. 870 - 892

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product v(K) = (Vol K)(Vol K°)...

52A53 (46B07, 53A05) | Mahler conjecture | Analysis | convex | Bourgain-Milman theorem | Gauss linking integral | Mathematics | Convex | MATHEMATICS | MINIMAL VOLUME-PRODUCT | BODIES | ZONOIDS

52A53 (46B07, 53A05) | Mahler conjecture | Analysis | convex | Bourgain-Milman theorem | Gauss linking integral | Mathematics | Convex | MATHEMATICS | MINIMAL VOLUME-PRODUCT | BODIES | ZONOIDS

Journal Article

Duke mathematical journal, ISSN 0012-7094, 2010, Volume 154, Issue 3, pp. 419 - 430

We prove that the unit cube B-infinity(n) is a strict local minimizer for the Mahler volume product vol(n)(K)vol(n)(K*) in the class of origin-symmetric convex...

MATHEMATICS | VOLUME-PRODUCT | BODIES | SPACES | ZONOIDS | Mathematics - Functional Analysis | 52A20 | 52A40

MATHEMATICS | VOLUME-PRODUCT | BODIES | SPACES | ZONOIDS | Mathematics - Functional Analysis | 52A20 | 52A40

Journal Article

Discrete & computational geometry, ISSN 1432-0444, 2019, Volume 62, Issue 3, pp. 583 - 600

For a convex body $$K \subset {\mathbb {R}}^n,$$ K ⊂ R n , let $$K^z = \{y\in {\mathbb R}^n : \langle y-z, x-z\rangle \le 1,\ \text{ for } \text{ all }\ x\in...

52A20 | Computational Mathematics and Numerical Analysis | 52A40 | Mahler conjecture | Volume product | Simplicial polytope | Mathematics | 52A38 | Combinatorics | Polar Bodies | MATHEMATICS | LOCAL MINIMALITY | INEQUALITIES | CONVEX-BODIES | BANACH-SPACES | COMPUTER SCIENCE, THEORY & METHODS | ZONOIDS | Polytopes | Polarity | Apexes | Combinatorial analysis

52A20 | Computational Mathematics and Numerical Analysis | 52A40 | Mahler conjecture | Volume product | Simplicial polytope | Mathematics | 52A38 | Combinatorics | Polar Bodies | MATHEMATICS | LOCAL MINIMALITY | INEQUALITIES | CONVEX-BODIES | BANACH-SPACES | COMPUTER SCIENCE, THEORY & METHODS | ZONOIDS | Polytopes | Polarity | Apexes | Combinatorial analysis

Journal Article

Advances in Mathematics, ISSN 0001-8708, 06/2013, Volume 240, pp. 613 - 635

Shadow systems are used to establish new asymmetric volume product and asymmetric volume ratio inequalities, along with their equality conditions. These...

Asymmetric | Shadow systems | Brunn-Minkowski theory | L-0-MINKOWSKI PROBLEM | MAHLER CONJECTURE | CONVEX-SETS | ZONOIDS | AFFINE ISOPERIMETRIC-INEQUALITIES | MATHEMATICS | Asymmetric L-p Brunn-Minkowski theory | PETTY CENTROID INEQUALITY | MINIMAL VOLUME-PRODUCT | MINKOWSKI-FIREY THEORY | VALUATIONS | BODIES

Asymmetric | Shadow systems | Brunn-Minkowski theory | L-0-MINKOWSKI PROBLEM | MAHLER CONJECTURE | CONVEX-SETS | ZONOIDS | AFFINE ISOPERIMETRIC-INEQUALITIES | MATHEMATICS | Asymmetric L-p Brunn-Minkowski theory | PETTY CENTROID INEQUALITY | MINIMAL VOLUME-PRODUCT | MINKOWSKI-FIREY THEORY | VALUATIONS | BODIES

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 07/2012, Volume 391, Issue 1, pp. 183 - 189

In this paper, we define the Orlicz zonotopes Zϕ. Using the notion of shadow systems, we give a sharp upper estimate for the volume of Zϕ and a sharp lower...

Shadow system | Mahler conjecture | Orlicz zonotopes | MATHEMATICS | MATHEMATICS, APPLIED | CENTROID INEQUALITY | CONVEX-BODIES | PRODUCT | SETS | ZONOIDS | L-P

Shadow system | Mahler conjecture | Orlicz zonotopes | MATHEMATICS | MATHEMATICS, APPLIED | CENTROID INEQUALITY | CONVEX-BODIES | PRODUCT | SETS | ZONOIDS | L-P

Journal Article

DUKE MATHEMATICAL JOURNAL, ISSN 0012-7094, 04/2020, Volume 169, Issue 6, pp. 1077 - 1134

We prove Mahler's conjecture concerning the volume product of centrally symmetric, convex bodies in R-n in the case where n = 3. More precisely, we show that,...

MATHEMATICS | LOCAL MINIMALITY | CONVEX-BODIES | ZONOIDS

MATHEMATICS | LOCAL MINIMALITY | CONVEX-BODIES | ZONOIDS

Journal Article

Advances in Mathematics, ISSN 0001-8708, 2011, Volume 228, Issue 5, pp. 2634 - 2646

We prove inequalities for mixed volumes of zonoids with isotropic generating measures. A special case is an inequality for zonoids that is reverse to the...

Zonoid | Reverse Urysohn inequality | Hyperplane process | Intersection density | Isotropic measure | Associated zonoid | Mixed volume | Characterization of parallelepipeds | MATHEMATICS | CONVEX-BODIES | SUBSPACES

Zonoid | Reverse Urysohn inequality | Hyperplane process | Intersection density | Isotropic measure | Associated zonoid | Mixed volume | Characterization of parallelepipeds | MATHEMATICS | CONVEX-BODIES | SUBSPACES

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 02/2014, Volume 266, Issue 4, pp. 2360 - 2402

Mahlerʼs conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in a fixed...

Polar bodies | Volume product | Hanner polytopes | Convex bodies | Mahlerʼs conjecture | Mahler's conjecture | LOCAL MINIMALITY | MAHLER CONJECTURE | PROOF | ZONOIDS | CONCAVE FUNCTIONS | INTEGRALS | MATHEMATICS | BANACH-SPACES | SANTALO INEQUALITY

Polar bodies | Volume product | Hanner polytopes | Convex bodies | Mahlerʼs conjecture | Mahler's conjecture | LOCAL MINIMALITY | MAHLER CONJECTURE | PROOF | ZONOIDS | CONCAVE FUNCTIONS | INTEGRALS | MATHEMATICS | BANACH-SPACES | SANTALO INEQUALITY

Journal Article

Advances in Mathematics, ISSN 0001-8708, 2010, Volume 225, Issue 4, pp. 1914 - 1928

A stability version of the Blaschke–Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n ⩾ 3 is proved. The first step...

Affine invariant inequalities | Stability | MATHEMATICS | FALSE CENTER | CONVEX-BODIES | MINIMAL VOLUME-PRODUCT | ELLIPSOIDS | SURFACE | ZONOIDS | Equality

Affine invariant inequalities | Stability | MATHEMATICS | FALSE CENTER | CONVEX-BODIES | MINIMAL VOLUME-PRODUCT | ELLIPSOIDS | SURFACE | ZONOIDS | Equality

Journal Article

Journal of the American Statistical Association, ISSN 0162-1459, 2019, Volume 115, Issue 529, pp. 1 - 24

We present single imputation method for missing values which borrows the idea of data depth-a measure of centrality defined for an arbitrary point of a space...

Outliers | Elliptical symmetry | Convex optimization | Nonparametric imputation | Zonoid depth | Tukey depth | Local depth | STATISTICS & PROBABILITY | Methodology | Statistics

Outliers | Elliptical symmetry | Convex optimization | Nonparametric imputation | Zonoid depth | Tukey depth | Local depth | STATISTICS & PROBABILITY | Methodology | Statistics

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 07/2016, Volume 368, Issue 7, pp. 5093 - 5124

, which corresponds to mean width, the inequality obtained confirms a conjecture of Betke and McMullen made in 1983.]]>

Betke-McMullen conjecture | Loomis-Whitney inequality | Zonoid | Cauchy-Binet theorem | Geometric tomography | Convex body | Intrinsic volume | Meyer’s inequality | CONVEX-BODIES | VOLUME | SUMSETS | geometric tomography | zonoid | MATHEMATICS | Meyer's inequality | SETS | PROJECTIONS | intrinsic volume | ENTROPY

Betke-McMullen conjecture | Loomis-Whitney inequality | Zonoid | Cauchy-Binet theorem | Geometric tomography | Convex body | Intrinsic volume | Meyer’s inequality | CONVEX-BODIES | VOLUME | SUMSETS | geometric tomography | zonoid | MATHEMATICS | Meyer's inequality | SETS | PROJECTIONS | intrinsic volume | ENTROPY

Journal Article

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