Zonoids with minimal volume-product

Reisner, Shlomo

doi:10.1007/bf01164009

Cited by 104 publications

(71 citation statements)

References 5 publications

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“…For instance, Saint-Raymond [19] observed that the Mahler volume of an n-cube C n is tied not only by the volume of the dual C * n , a cross-polytope, but also by other polytopes when n is large. (Nonetheless, the Mahler conjecture has been established in some special cases, including 1-unconditional convex bodies [19] and zonoids [9,17]. )…”

Section: Conjecture 12 (Mahler) In Any Fixed Dimension N V(k) Is Mmentioning

confidence: 99%

From the Mahler Conjecture to Gauss Linking Integrals

Kuperberg

2008

GAFA Geom. funct. anal.

153

102

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Dedicated to my father, on no particular occasionWe 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 • ) of the volume of a symmetric convex body K ∈ R n and its polar body K • . The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domainwhere γ n is a monotonic factor that begins at 4 π and converges to √ 2. This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of c n .The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K ♦ , with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere S n−1 and in hyperbolic space H n−1 .

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Section: Conjecture 12 (Mahler) In Any Fixed Dimension N V(k) Is Mmentioning

confidence: 99%

From the Mahler Conjecture to Gauss Linking Integrals

Kuperberg

2008

GAFA Geom. funct. anal.

153

102

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“…Since the appearance of Bolker's article, projection bodies have received considerable increased attention (see, for example, [7,13,14,21,23,26,27,28,32,40, 41] Mixed projection bodies are related to ordinary projection bodies in the same way that mixed volumes are related to ordinary volume. The definition and some elementary properties of mixed projection bodies can be found in the classic volume of .…”

mentioning

confidence: 99%

“…Since the appearance of Bolker's article, projection bodies have received considerable increased attention (see, for example, [7,13,14,21,23,26,27,28,32,40,41]). Also, new applications have appeared in combinatorics (see Stanley [36]), in stereology (see ), in stochastic geometry (see Schneider [32,33]), in mathematical economics (see Hildenbrand [16]), and even in the study of random determinants (see Vitale [39]).…”

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confidence: 99%

Inequalities for mixed projection bodies

Lutwak¹

1993

Trans. Amer. Math. Soc.

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Abstract. Mixed projection bodies are related to ordinary projection bodies (zonoids) in the same way that mixed volumes are related to ordinary volume. Analogs of the classical inequalities from the Brunn-Minkowski Theory (such as the Minkowski, Brunn-Minkowski, and Aleksandrov-Fenchel inequalities) are developed for projection and mixed projection bodies.Two decades ago, Bolker [5] observed that projection bodies (also known as zonoids) were objects of independent investigation in a number of mathematical disciplines such as measure theory, crystallography, optimal control theory, functional analysis, and geometric convexity. Since the appearance of Bolker's article, projection bodies have received considerable increased attention (see, for example, [7,13,14,21,23,26,27,28,32,40, 41] Mixed projection bodies are related to ordinary projection bodies in the same way that mixed volumes are related to ordinary volume. The definition and some elementary properties of mixed projection bodies can be found in the classic volume of . Support functions of mixed projection bodies were investigated by Chakerian [9]. Stability questions for mixed projection bodies are treated by Goodey [11] and . In [18] and [19], inequalities for the polars of mixed projection bodies were obtained. This article treats the corresponding inequalities for the mixed projection bodies themselves. Analogs of the classical mixed volume inequalities (such as the Brunn-Minkowski, Minkowski, and Aleksandrov-Fenchel inequalities) will be established for mixed projection bodies.Since interest in zonoids is not limited to one discipline, an attempt is made to make this article reasonably self-contained.Background material and notation regarding mixed volumes and mixed surface area measures is given in §0. The classical mixed volume inequalities are

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“…Based on the above proof of the Lemma 2.1, and using the reverse Blaschke-Santaló inequality (see [5,25]),…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Dual Orlicz mixed geominimal surface area

Gao¹,

Ma²,

Guo³

2018

J. Nonlinear Sci. Appl.

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Based on the classical ideas, Zhu discussed the properties and useful theories for L p -mixed geominimal surface area; meanwhile, Ma defined dual Orlicz geominimal surface area. The previous studies provide a thought to us for the study of the dual Orlicz mixed geominimal surface. In our paper, we have done the following work: attempting to use an integral form to define the dual Orlicz mixed geominimal surface area, further studyding its related properties, and listing some inequalities including Alexandrov-Fenchel type inequality, analogous cyclic inequality, Blaschke-Santaló type inequality, and affine isoperimetric inequality in Orlicz space.

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Zonoids with minimal volume-product

Cited by 104 publications

References 5 publications

From the Mahler Conjecture to Gauss Linking Integrals

From the Mahler Conjecture to Gauss Linking Integrals

Inequalities for mixed projection bodies

Dual Orlicz mixed geominimal surface area

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