The floating body in real space forms

Besau, Florian; Werner, Elisabeth M.

doi:10.4310/jdg/1538791243

Cited by 29 publications

(35 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some results in the spirit of those discussed in Section 5 can be obtained also for spherical convex floating bodies [14]. In another paper, Besau and Werner [15] provide extensions of those results also to certain Riemannian manifolds. Research in this direction in the statistics of data depth is still only in its beginnings [55].…”

Section: Conclusion and Further Perspectivesmentioning

confidence: 70%

See 1 more Smart Citation

Halfspace depth and floating body

Nagy¹,

Schütt²,

Werner³

2019

Statist. Surv.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Conclusion and Further Perspectivesmentioning

confidence: 70%

“…Proposition 14. Let K ∈ K d , and let δ 0 be as in (15). Then vol d (K \ K δ ) is a differentiable function of δ on (0, δ 0 ) and…”

Section: 2mentioning

confidence: 99%

Halfspace depth and floating body

Nagy¹,

Schütt²,

Werner³

2019

Statist. Surv.

Self Cite

View full text Add to dashboard Cite

show abstract

“…the books by R. Gardner [16] and R. Schneider [46] and also [30,33,35,52,53,54,58]. Recent developments include extensions to an Orlicz theory, e.g., [17,27,33,59], to a functional setting [12,13] and to the spherical and hyperbolic setting [5,6]. Applications of affine surface areas have been manifold.…”

Section: Introductionmentioning

confidence: 99%

Constrained convex bodies with extremal affine surface areas

Giladi

Huang

Schütt

et al. 2020

Journal of Functional Analysis

View full text Add to dashboard Cite

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. *

show abstract

“…Our first theorem shows that (weighted) Ulam's floating bodies are isomorphic, in a sense, to (weighted) floating bodies. Weighted floating bodies were introduced in (also see for recent applications) as follows. Let

K \subseteq {double-struckR}^{n}

be a convex body,

0 < δ

, and

ϕ : K \to R

be integrable and such that

ϕ > 0

almost everywhere with respect to Lebesgue measure.…”

Section: Introductionmentioning

confidence: 99%

Ulam floating bodies

Huang

Slomka

Werner

2019

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

We study a new construction of bodies from a given convex body in Rn which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to p‐affine surface areas. We show that these bodies are related to Ulam' s long‐standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.

show abstract

The floating body in real space forms

Cited by 29 publications

References 48 publications

Halfspace depth and floating body

Halfspace depth and floating body

Constrained convex bodies with extremal affine surface areas

Ulam floating bodies

Contact Info

Product

Resources

About