Related papers: Weighted floating bodies and polytopal approximati…
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 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 derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has…
Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to…
Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and…
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…
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 define new surface area measures for ball-convex bodies which we call $L_p$ relative surface areas. We show that those are rigid motion invariant valuations. We establish inequalities for these quantities and prove a monotonicity…
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…
We introduce the flag-approximability of a convex body to measure how easy it is to approximate by polytopes. We show that the flag-approximability is exactly half the volume entropy of the Hilbert geometry on the body, and that both…
The paper is devoted to the study of asymptotic behavior of solutions for nonlocal elliptic problems in weighted spaces. We deal with the most difficult case where the support of nonlocal terms intersects with the boundary of a plane…
The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three…
A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes…
An asymptotic formula is proved for the expected $T$-functional of the convex hull of independent and identically distributed random points sampled from the Euclidean unit sphere in $\mathbb{R}^n$ according to an arbitrary positive…
We show that the spheres in Hilbert geometry have the same volume growth entropy as those in the Lobachevsky space. We give the asymptotic estimates for the ratio of the volume of metric ball to the area of the metric sphere in Hilbert…
The floating structure problem describes the interaction between surface water waves and a floating body, generally a boat or a wave energy converter. As shown by Lannes in [18] the equations for the fluid motion can be reduced to a set of…
We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their…
We observe that a large part of the volume of a hyperbolic polyhedron is taken by a tubular neighbourhood of its boundary, and use this to give a new proof for the finiteness of arithmetic maximal reflection groups following a recent work…
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.
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…