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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Journal Article
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