Asymptotic Geometric Analysis, Part I

Artstein-Avidan, Shiri; Giannopoulos, Apostolos; Milman, Vitali

doi:10.1090/surv/202

Cited by 235 publications

(334 citation statements)

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“…They are related to the isotropic position, to the study of volume concentration, volume ratio, reverse isoperimetric inequalities, Banach-Mazur distance of normed spaces, and many more, including the hyperplane conjecture, one of the major open problems in asymptotic geometric analysis. We refer to e.g., the books [1,11] for the details and more information.…”

Section: Introductionmentioning

confidence: 99%

Constrained convex bodies with extremal affine surface areas

Giladi

Huang

Schütt

et al. 2020

Journal of Functional Analysis

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Given a convex body K ⊆ R n and p ∈ R, we introduce and study the extremal inner and outer affine surface areaswhere asp(K ′ ) denotes the Lp-affine surface area of K ′ , and the supremum is taken over all convex subsets of K and the infimum over all convex compact subsets containing K. The convex body that realizes IS1(K) in dimension 2 was determined in [3] where it was also shown that this body is the limit shape of lattice polytopes in K. In higher dimensions no results are known about the extremal bodies. We use a thin shell estimate of [22] and the Löwner ellipsoid to give asymptotic estimates on the size of ISp(K) and osp (K). Surprisingly, it turns out that both quantities are proportional to a power of volume. *

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Section: Introductionmentioning

confidence: 99%

Constrained convex bodies with extremal affine surface areas

Giladi

Huang

Schütt

et al. 2020

Journal of Functional Analysis

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“…Proof. (1). Since ⊗ π corresponds, under the bijection given in Theorem 3.8, to the projective norm π(·), then, by Proposition 3.9, ⊗ π inj corresponds to the biggest injective tensor norm /π \.…”

Section: 4mentioning

confidence: 97%

A general theory of tensor products of convex sets in Euclidean spaces

Fernández-Unzueta

Higueras-Montaño

2020

Positivity

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We introduce both the notions of tensor product of convex bodies that contain zero in the interior, and of tensor product of 0-symmetric convex bodies in Euclidean spaces. We prove that there is a bijection between tensor products of 0-symmetric convex bodies and tensor norms on finite dimensional spaces. This bijection preserves duality, injectivity and projectivity. We obtain a formulation of Grothendieck's Theorem for 0-symmetric convex bodies and use it to give a geometric representation (up to the KG-constant) of the Hilbertian tensor product. We see that the property of having enough symmetries is preserved by these tensor products, and exhibit relations with the Löwner and the John ellipsoids.

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“…We refer to the book of Schneider [29] for basic facts from Brunn–Minkowski theory and to the book of Artstein‐Avidan et al [2] for basic facts from asymptotic convex geometry.…”

Section: Notation and Backgroundmentioning

confidence: 99%

“…We say that

K

is symmetric if

x \in K

implies that

- x \in K

, and that

K

is centered if its barycenter

\begin{matrix} true \\ right bar (K) = \frac{1}{{vol}_{n} (K)} \int_{K} x d x \end{matrix}

is at the origin. The polar body

K^{\circ}

K

is defined by

\begin{matrix} true \\ right K^{\circ} : = {y \in {double-struckR}^{n} : 〈 x, y 〉 ⩽ 1 for all x \in K} . \end{matrix}

The Blaschke–Santaló inequality (see [2, Theorem 1.5.10]) states that for every centered convex body

K

R^{n}

, one has

{vol}_{n} (K) {vol}_{n} (K^{\circ}) ⩽ ω_{n}^{2}

, with equality if and only if

K

is an ellipsoid. The reverse Santaló inequality of Bourgain and Milman [10] states that there exists an absolute constant

c > 0

such that

\begin{matrix} true \\ right {({vol}_{n} (K) {vol}_{n} (K^{\circ}))}^{1 / n} ⩾ c / n \end{matrix}

for every convex body

K

R^{n}

which contains

0

in its interior.…”

Section: Notation and Backgroundmentioning

confidence: 99%

Brascamp–lieb Inequality and Quantitative Versions of Helly's Theorem

Brazitikos

2016

Mathematika

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We provide new quantitative versions of Helly's theorem. For example, we show that for every family {Pi:i∈I} of closed half‐spaces in Rn such that P=⋂i∈IPi has positive volume, there exist s⩽αn and i1,…,is∈I such that right center left3ptthickmathspacetrueitalicvolMJX-TeXAtom-ORDn⁡(PMJX-TeXAtom-ORDiMJX-TeXAtom-ORD1∩⋯∩PMJX-TeXAtom-ORDiMJX-TeXAtom-ORDs)⩽(Cn) MJX-TeXAtom-ORDnitalicvolMJX-TeXAtom-ORDn⁡(P), where α,C>0 are absolute constants. These results complement and improve previous work of Bárány et al and Naszódi. Our method combines the work of Srivastava on approximate John's decompositions with few vectors, a new estimate on the corresponding constant in the Brascamp–Lieb inequality and an appropriate variant of Ball's proof of the reverse isoperimetric inequality.

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Asymptotic Geometric Analysis, Part I

Cited by 235 publications

References 0 publications

Constrained convex bodies with extremal affine surface areas

Constrained convex bodies with extremal affine surface areas

A general theory of tensor products of convex sets in Euclidean spaces

Brascamp–lieb Inequality and Quantitative Versions of Helly's Theorem

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