Geometry of Isotropic Convex Bodies

Brazitikos, Silouanos

doi:10.1090/surv/196

Cited by 224 publications

(280 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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“…A typical example of a log-concave vector is a vector uniformly distributed over a convex body. Various results and conjectures about log-concave measures are discussed in the recently published monograph [4].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Weak and strong moments of ℓᵣ-norms of log-concave vectors

Latała¹,

Strzelecka²

2016

Proc. Amer. Math. Soc.

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Weak and strong moments of ℓ r -norms of log-concave vectors * Rafa l Lata la and Marta Strzelecka revised version Abstract We show that for p ≥ 1 and r ≥ 1 the p-th moment of the ℓ r -norm of a logconcave random vector is comparable to the sum of the first moment and the weak p-th moment up to a constant proportional to r. This extends the previous result of Paouris concerning Euclidean norms.

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“…with ϕ : [0, ∞) → [0, ∞). In the special case of exponential Orlicz norms · ψα such an equivalence is a classical result in asymptotic geometric analysis and may be found, without explicit constants, in the monographs [3, Lemma 3.5.5] and [4,Lemma 2.4.2].…”

Section: The Proofsmentioning

confidence: 99%