Comparisons of best and random approximation of convex bodies by polytopes. (English) Zbl 0896.52014
Stoka, Marius I. (ed.), Second international conference in stochastic geometry, convex bodies and empirical measures, Agrigento, Italy, September 9–14, 1996. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 50, 189-216 (1997).
This is an interesting survey on results concerning the approximation of a convex body \(C\) by polytopes. The author compares the behaviour of best approximating polytopes with at most \(n\) vertices (facets) with the one of the convex hull of \(n\) random points in \(C\) (random half spaces containing \(C)\). Both situations are considered, the case of fixed \(n\) and the asymptotic case, as \(n\to \infty\).
The paper is divided in three parts. The most (and most precise) results are given for the symmetric difference metric (deviation in volume). Here, also variants for other quermassintegrals are treated. Rather less seems to be known on the approximations in the Hausdorff metric or (in the symmetric case) in the Banach-Mazur distance. The final part gives a few results on floating bodies and expectations of random polytopes.
For the entire collection see [Zbl 0878.00066].
The paper is divided in three parts. The most (and most precise) results are given for the symmetric difference metric (deviation in volume). Here, also variants for other quermassintegrals are treated. Rather less seems to be known on the approximations in the Hausdorff metric or (in the symmetric case) in the Banach-Mazur distance. The final part gives a few results on floating bodies and expectations of random polytopes.
For the entire collection see [Zbl 0878.00066].
Reviewer: W.Weil (Karlsruhe)
MSC:
| 52A22 | Random convex sets and integral geometry (aspects of convex geometry) |
| 52A27 | Approximation by convex sets |
| 60D05 | Geometric probability and stochastic geometry |
