Weighted floating bodies and polytopal approximation

Abstract

Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floating areas are combined to derive new asymptotic approximation results on the sphere, in hyperbolic space and in Hilbert geometries.2000 AMS subject classification: Primary 52A38; Secondary 52A27, 52A55, 53C60, 60D05.Let K be a convex body (that is, compact convex set) in R n . For δ > 0, the floating body K δ of K is obtained by cutting off all caps that have volume less or equal to δ. Extending results for smooth bodies (cf. [18]), Schütt and Werner [34] showed for a general convex bodywhere α n is an explicitly known positive constant (see Section 1.1). Here V n is n-dimensional volume, H n−1 (K, x) is the Gauss-Kronecker curvature at x ∈ ∂K and integration is with respect to the (n − 1)-dimensional Hausdorff measure. The integral on the right side is the affine surface area of K (cf. [20,22] and [31, Section 10.5] for more information). Affine surface area also determines the asymptotic behavior of random polytopes. Specifically, choose m points uniformly and independently in K and denote their convex hull by K m . The random polytope K m is easily seen to converge to K in the sense that E(V n (K) − V n (K m )) → 0 as m → ∞, where E denotes expectation. The asymptotic behavior of K m has been studied extensively since the 1960's, starting with the seminal results by Rényi and Sulanke [28, 29] (cf. [17, 27]). Extending results of Bárány [1], Schütt [33] was able to prove the analog to (1) for the random polytope K m in a general convex body K,

Flag numbers and floating bodies

“…for every sufficiently small δ > 0 (cf. [15,Eq. (3)]). Clearly, if ϕ ≡ 1, then the weighted floating body is the convex floating body, i.e., P ϕ δ = P δ .…”

Random inscribed polytopes in projective geometries

“…The proofs of (2.1) and (2.2) for the unweighted case (i.e., when is the Lebesgue measure and σ is the (d − 1)-dimensional Hausdorff measure) can be extracted from existing literature (see [65,Lemma 6.3] and [57, Lemma 6.2]), and the general case follows by a 'sandwiching' argument similar to [8,Lemma 5.2]. For transparency, we sketch a direct argument below.…”

“…Following the ideas of [8], we reformulate the problem in terms of weighted random inscribed polytopes in Euclidean space. This approach is more general and also paves the way to some new directions.…”

Ulam floating bodies

“…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 be a convex body, , and be integrable and such that almost everywhere with respect to Lebesgue measure.…”