On some random convex sets

Some results on functionally convex sets in real Banach spaces

‎We use of two notions functionally convex (briefly‎, ‎F--convex) and functionally closed (briefly‎, ‎F--closed) in functional analysis and obtain more results‎. ‎We show that if $lbrace A_{alpha} rbrace _{alpha in I}$ is a family $F$--convex subsets with non empty intersection of a Banach space $X$‎, ‎then $bigcup_{alphain I}A_{alpha}$ is F--convex‎. ‎Moreover‎, ‎we introduce new definition o...

Large convex holes in random point sets

A convex hole (or empty convex polygon) of a point set P in the plane is a convex polygon with vertices in P , containing no points of P in its interior. Let R be a bounded convex region in the plane. We show that the expected number of vertices of the largest convex hole of a set of n random points chosen independently and uniformly over R is Θ(logn/(log log n)), regardless of the shape of R.

Random generation of Q-convex sets

The problem of randomly generating Q-convex sets is considered. We present two generators. The first one uses the Q-convex hull of a set of random points in order to generate a Q-convex set included in the square [0, n)2. This generator is very simple, but is not uniform and its performance is quadratic in n. The second one exploits a coding of the salient points, which generalizes the coding o...

On the hyperplane conjecture for random convex sets

This seemingly innocuous question, considered two decades ago by Bourgain [3, 4], has not been answered yet. We refer the reader to, e.g., [2], [18] or [14] for partial results, history and for additional literature regarding this hyperplane conjecture. In particular, there are large classes of convex bodies for which an affirmative answer to the above question is known. These include unconditi...

Random Walks on Nite Convex Sets of Lattice Points