Section: New Results

Classical and probabilistic computational geometry

Participants : Sylvain Lazard, Marc Pouget.

Worst-case silhouette size of random polytopes

We studied from a probabilistic point of view the size of the silhouette of a polyhedron. While the silhouette size of a polyhedron with $n$ vertices may be linear for some view points, several experimental and theoretical studies show a sublinear behavior for a wide range of constraints. The latest result on the subject proves a bound in $\Theta \left(\sqrt{n}\right)$ on the size of the silhouette from a random view point of polyhedra of size $n$ approximating non-convex surfaces in a reasonable way [7] . In this result, the polyhedron is considered given and the sizes of its silhouettes are averaged over all view points. We addressed the problem of bounding the worst-case size of the silhouette where the average is taken over a set of polyhedra. Namely, we consider random polytopes defined as the convex hull of a Poisson point process on a sphere in ${ℝ}^3$ such that its average number of points is $n$. We show that the expectation over all such random polytopes of the maximum size of their silhouettes viewed from infinity is $\Theta \left(\sqrt{n}\right)$. This work, done in collaboration with Marc Glisse (Inria Geometrica) and Julien Michel (Université de Poitiers), was submitted this year to the Journal of Computational Geometry [28] .

Recognizing shrinkable complexes is NP-complete

We say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove [14] that it is NP-complete to decide whether a (three-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes which are not shrinkable. This work was done in collaboration with Dominique Attali (CNRS, Grenoble), Olivier Devillers and Marc Glisse (Inria Geometrica).

On point-sets that support planar graphs

A set of points is said universal if it supports a crossing-free drawing of any planar graph. For a planar graph with $n$ vertices, if edges can be drawn as polylines with at most one bend, we exhibited universal point-sets of size $n$ if the bend-points can be placed arbitrarily [26] . Furthermore, if the bend points are also required to be chosen in the universal set, we proved the existence of universal sets of subquadratic size, $O(n^2/logn)$ [25] . More recently, we considered the setting in which graphs are drawn with curved edges. We proved that, surprisingly, there also exists a universal set of $n$ points in the plane for which every $n$-vertex planar graph admits a planar drawing in which the edges are drawn as a circular arc [11] .