Symmetric Mahler’s conjecture for the volume product in the 3-dimensional case

Iriyeh, Hiroshi; Shibata, Masataka

doi:10.1215/00127094-2019-0072

Cited by 45 publications

(30 citation statements)

References 23 publications

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“…In the case the body K is centrally symmetric, Mahler [31] conjectured that the minimizer is the unit cube, and he proved it in [30] for n = 2. Recently, the symmetric case was also proved in the affirmative for n = 3 by Iriyeh and Shibata [24]. In general dimensions, it was shown by Nazarov et.…”

Section: The Mahler Volume Product Of Polytopesmentioning

confidence: 80%

A Concentration Inequality for Random Polytopes, Dirichlet–Voronoi Tiling Numbers and the Geometric Balls and Bins Problem

Hoehner

Kur

2020

Discrete Comput Geom

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Our main contribution is a concentration inequality for the symmetric volume difference of a C 2 convex body with positive Gaussian curvature and a circumscribed random polytope with a restricted number of facets, for any probability measure on the boundary with a positive density function.We also show that the Dirichlet-Voronoi tiling numbers satisfy divn−1 = (2πe) −1 (n + ln n) + O(1), which improves a classical result of Zador by a factor of o(n). In addition, we provide a remarkable open problem which is the natural geometric generalization of the famous and fundamental "balls and bins" problem from probability. This problem is tightly connected to the optimality of random polytopes in high dimensions.Finally, as an application of the aforementioned results, we derive a lower bound for the maximal Mahler volume product of polytopes with a restricted number of vertices or facets.

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Section: The Mahler Volume Product Of Polytopesmentioning

confidence: 80%

A Concentration Inequality for Random Polytopes, Dirichlet–Voronoi Tiling Numbers and the Geometric Balls and Bins Problem

Hoehner

Kur

2020

Discrete Comput Geom

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“…The famous Mahler conjecture states that the volume product is minimized for affine transforms of cubes (among others) and a proof for the two-dimensional case is due to Mahler [33]. More recently, the conjecture was confirmed for the three-dimensional case [23], but the general case remains open.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Invariant Valuations on Super-Coercive Convex Functions

Mussnig¹

2019

Can. J. Math.-J. Can. Math.

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All non-negative, continuous, SL(n) and translation invariant valuations on the space of super-coercive, convex functions on R n are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, SL(n) and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume and polar volume are characterized.2010 AMS subject classification: 26B25 (46A40, 52A20, 52A41, 52B45).

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“…The Mahler conjecture, one of the central questions in convex geometry, asserts that for any origin-symmetric convex body L, the Mahler volume Vol n (L)·Vol n (L • ) is greater or equal to the Mahler volume of the cube Vol n (B n ∞ )·Vol n (B n 1 ), where L • denotes the polar body for L. We refer to [5] (see also [11]) for an "isomorphic solution" to this problem and related information. Very recently, this conjecture was verified in R 3 in [10]. In [17] the Mahler conjecture was confirmed in every dimension in a small neighborhood of the cube and, moreover, it was shown that the cube is a strict local minimizer in the Banach-Mazur metric on the class of symmetric convex bodies.…”

Section: Introductionmentioning

confidence: 85%

Cube is a Strict Local Maximizer for the Illumination Number

Livshyts

Tikhomirov

2019

Discrete Comput Geom

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It was conjectured by Levi, Hadwiger, Gohberg and Markus that the boundary of any convex body in R n can be illuminated by at most 2 n light sources, and, moreover, 2 n − 1 light sources suffice unless the body is a parallelotope. We show that if a convex body is close to the cube in the Banach-Mazur metric, and it is not a parallelotope, then indeed 2 n − 1 light sources suffice to illuminate its boundary. Equivalently, any convex body sufficiently close to the cube, but not isometric to it, can be covered by 2 n − 1 smaller homothetic copies of itself.2010 Mathematics Subject Classification. 52A20, 52C17.

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Symmetric Mahler’s conjecture for the volume product in the 3-dimensional case

Cited by 45 publications

References 23 publications

A Concentration Inequality for Random Polytopes, Dirichlet–Voronoi Tiling Numbers and the Geometric Balls and Bins Problem

A Concentration Inequality for Random Polytopes, Dirichlet–Voronoi Tiling Numbers and the Geometric Balls and Bins Problem

Invariant Valuations on Super-Coercive Convex Functions

Cube is a Strict Local Maximizer for the Illumination Number

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