Related papers: Floating and Illumination Bodies for Polytopes: Du…

We investigate a duality relation between floating and illumination bodies. The definitions of these two bodies suggest that the polar of the floating body should be similar to the illumination body of the polar. We consider this question…

Let $K$ be a convex body in $\Bbb R^{d}$ and $K_{t}$ its floating bodies. There is a polytope with at most $n$ vertices that satisfies $$ K_{t} \subset P_{n} \subset K $$ where $$ n \leq e^{16d} \frac{vol_{d}(K \setminus K_{t})}{t\…

The concept of illumination bodies studied in convex geometry is used to amend the halfspace depth for multivariate data. The proposed notion of illumination enables finer resolution of the sample points, naturally breaks ties in the…

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…

We define floating bodies in the class of $n$-dimensional ball-convex bodies. A right derivative of volume of these floating bodies leads to a surface area measure for ball-convex bodies which we call relative affine surface area. We show…

In this paper, we give an overview of some results concerning best and random approximation of convex bodies by polytopes. We explain how both are linked and see that random approximation is almost as good as best approximation.

While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex…

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and…

We introduce floating bodies for convex, not necessarily bounded subsets of $\mathbb{R}^n$. This allows us to define floating functions for convex and log concave functions and log concave measures. We establish the asymptotic behavior of…

We investigate several closely related "homothety conjectures" for convex bodies on a plane. Using the modern language of differential geometry, we systematically derive the fundamental properties of bodies of flotation, bodies of buoyancy,…

Three-dimensional central symmetric bodies different from spheres that can float in all orientations are considered. For relative density rho=1/2 there are solutions, if holes in the body are allowed. For rho different from 1/2 the body is…

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…

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in R^n by its dual L_{p$-centroid bodies is independent of the geometry of P. In particular we show that if P has volume 1,…

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.…

It is proved that every convex body in the plane has a point such that the union of the body and its image under reflection in the point is convex. If the body is not centrally symmetric, then it has, in fact, three affinely independent…

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar…

For a pseudoconvex tube domain, we prove estimates that relate the sublevel sets of its diagonal Bergman kernel to the floating bodies of its convex base. This allows us to associate a new affine invariant to any convex body.

The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Fefferman's invariant hypersurface measure.

The distance between convex bodies \(K, L \subseteq \R^n\) is defined as \[ d(K,L)= \inf \left\{ \lambda \ge 1: \ L-x \subseteq T (K-y) \subseteq \lambda (L-x) \right\}, \] where the infimum is taken over all \(x,y \in \R^n\) and all…

We prove new $L_p$ affine isoperimetric inequalities for all $ p \in [-\infty,1)$. We establish, for all $p\neq -n$, a duality formula which shows that $L_p$ affine surface area of a convex body $K$ equals $L_\frac{n^2}{p}$ affine surface…