For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry. Several properties of the floating area are established.

We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space.Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies.

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,

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope that is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants. Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak’s dual Brunn–Minkowski theory.

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