Hausdorff approximation of 3D convex polytopes

Approximation properties of random polytopes associated with Poisson hyperplane processes

We consider a stationary Poisson hyperplane process with given directional distribution and intensity in d-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body K and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing K. We study how well these random polytopes approximate K (measured by the Hausdorff...

Computational Aspects of the Hausdorff Distance in Unbounded Dimension

We study the computational complexity of determining the Hausdorff distance of two polytopes given in halfspaceor vertex-presentation in arbitrary dimension. Subsequently, a matching problem is investigated where a convex body is allowed to be homothetically transformed in order to minimize its Hausdorff distance to another one. For this problem, we characterize optimal solutions, deduce a Hell...

The Lower Bound Theorem for polytopes that approximate C-convex bodies

The face numbers of simplicial polytopes that approximate C-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence {Pn}n=0 of simplicial polytopes converges to a C-convex body in the ...

Linear time approximation of 3D convex polytopes

Asymptotic approximation of smooth convex bodies by general polytopes

For the optimal approximation of convex bodies by inscribed or circumscribed polytopes there are precise asymptotic results with respect to different notions of distance. In this paper we want to derive some results on optimal approximation without restricting the polytopes to be inscribed or circumscribed. Let Pn and P(n) denote the set of polytopes with at most n vertices and n facets, respec...