Faster ASV decomposition for orthogonal polyhedra using the Extreme Vertices Model (EVM)

The alternating sum of volumes (ASV) decomposition is a widely used technique for converting a B-Rep into a CSG model. The obtained CSG tree has convex primitives at its leaf nodes, while the contents of its internal nodes alternate between the set union and difference operators. This work first shows that the obtained CSG tree T can also be expressed as the regularized Exclusive-OR operation among all the convex primitives at the leaf nodes of T, regardless the structure and internal nodes of T. This is an important result in the case in which EVM represented orthogonal polyhedra are used because in this model the Exclusive-OR operation runs much faster than set union and difference operations. Therefore this work applies this result to EVM represented orthogonal polyhedra. It also presents experimental results that corroborate the theoretical results and includes some practical uses for the ASV decomposition of orthogonal polyhedra.Postprint (published version

On Convex Envelopes and Regularization of Non-Convex Functionals without moving Global Minima

We provide theory for the computation of convex envelopes of non-convex functionals including an l2-term, and use these to suggest a method for regularizing a more general set of problems. The applications are particularly aimed at compressed sensing and low rank recovery problems but the theory relies on results which potentially could be useful also for other types of non-convex problems. For optimization problems where the l2-term contains a singular matrix we prove that the regularizations never move the global minima. This result in turn relies on a theorem concerning the structure of convex envelopes which is interesting in its own right. It says that at any point where the convex envelope does not touch the non-convex functional we necessarily have a direction in which the convex envelope is affine.Comment: arXiv admin note: text overlap with arXiv:1609.0937

Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting

Basic Polyhedral Theory

This is a chapter (planned to appear in Wiley's upcoming Encyclopedia of Operations Research and Management Science) describing parts of the theory of convex polyhedra that are particularly important for optimization. The topics include polyhedral and finitely generated cones, the Weyl-Minkowski Theorem, faces of polyhedra, projections of polyhedra, integral polyhedra, total dual integrality, and total unimodularity.Comment: 14 page