Ulam floating bodies

Huang, Han; Slomka, Boaz A.; Werner, Elisabeth M.

doi:10.1112/jlms.12226

Cited by 21 publications

(32 citation statements)

References 59 publications

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“…The expressions in the above theorem can thus be used to define the affine surface area for all convex bodies. Around the same time, different extensions of the affine surface area to arbitrary convex bodies were given by Leichtweiß [92] and Lutwak [105] and afterwards several more have been found, e.g., [81,124,178]. It has been shown that all those extensions coincide.…”

Section: Affine Surface Area An Important Affine Invariant From Affimentioning

confidence: 98%

Halfspace depth and floating body

Nagy¹,

Schütt²,

Werner³

2019

Statist. Surv.

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Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.

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Section: Affine Surface Area An Important Affine Invariant From Affimentioning

confidence: 98%

Halfspace depth and floating body

Nagy¹,

Schütt²,

Werner³

2019

Statist. Surv.

Self Cite

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“…Geometric descriptions in the sense of (2.6) and (2.7) of L p -affine surface area also exist. We refer to e.g., [28,50,51,56,58].…”

Section: Background and Definitionsmentioning

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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“…In special cases, our construction yields L p -centroid bodies (see [37] and [47,Sec. 10.8]) and Ulam floating bodies recently introduced in [27]. The latter form a particularly important special setting, which is confirmed by showing that all transformations K → E e (K) can be expressed in terms of Ulam floating bodies.…”

Section: Introductionmentioning

confidence: 73%

“…The following result establishes relationships between average quantile sets (or metronoids) and depth-trimmed regions. Its second part generalises [27,Th. 1.1], which concerns the case of ξ supported by a convex body.…”

Section: Average Quantile Sets As Integrated Depth-trimmed Regionsmentioning

confidence: 97%

Convex bodies generated by sublinear expectations of random vectors

Molchanov

Turin

2021

Advances in Applied Mathematics

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We show that many well-known transforms in convex geometry (in particular, centroid body, convex floating body, and Ulam floating body) are special instances of a general construction, relying on applying sublinear expectations to random vectors in Euclidean space. We identify the dual representation of such convex bodies and describe a construction that serves as a building block for all so defined convex bodies. Sublinear expectations are studied in mathematical finance within the theory of risk measures. In this way, tools from mathematical finance yield a whole variety of new geometric constructions.

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Ulam floating bodies

Cited by 21 publications

References 59 publications

Halfspace depth and floating body

Halfspace depth and floating body

Constrained convex bodies with extremal affine surface areas

Convex bodies generated by sublinear expectations of random vectors

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