THOMASSE , Rémy

In this thesis, we give some new results about the average size of convex hulls made of points chosen in a convex body. This size is known when the points are chosen uniformly (and independently) in a convex polytope or in a "smooth" enough convex body. This average size is also known if the points are independently chosen according to a...

The asymptotic behavior of the expected size of the convex hull of uniformly random points in a convex body in Rd is polynomial for a smooth body and polylogarithmic for a polytope. We construct a body whose expected size of the convex hull oscillates between these two behaviors when the number of points increases.

We propose an algorithm that generates a random polygon as a convex hull of n points uniformly and independently distributed in a disc without explicitly generate all the points.

We propose an algorithm that generates a random polygon as a convex hull of n points uniformly and independently distributed in a disc without explicitly generate all the points.

The asymptotic behavior of the size of the convex hull of uniformly random points in a convex body in Rd is known for polytopes and smooth convex bodies. These are the lower and the upper bound for a general convex body. In this paper, we exhibit an example of convex body whose size of the random convex hull alternates behavior close to the...

We present a simple technique for analyzing the size of geometric hypergraphs defined by random point sets. As an application we obtain upper and lower bounds on the smoothed number of faces of the convex hull under Euclidean and Gaussian noise and related results.

We establish an upper bound on the smoothed complexity of convex hulls in $\mathbb{R}^d$ under uniform Euclidean ($\ell^2$) noise. Specifically, let $\{p_1^*, p_2^*, \ldots, p_n^*\}$ be an arbitrary set of $n$ points in the unit ball in $\mathbb{R}^d$ and let $p_i=p_i^*+x_i$, where $x_1, x_2, \ldots, x_n$ are chosen independently from the...

We present a simple technique for analyzing the size of geometric hypergraphs dened by random point sets. As an application we obtain upper and lower bounds on the smoothed number of faces of the convex hull under Euclidean and Gaussian noise and related results.