Let S be a finite set of n points in the plane in general position. We prove that every inclusion-maximal family of subsets of S separable by convex pseudo-circles has the same cardinal (n 0)+(n 1)+(n 2)+(n 3). This number does not depend on the configuration of S and is the same as the number of subsets of S separable by true circles. Buzaglo, Holzman, and Pinchasi already showed that it is an upper bound for the number of subsets separable by (non necessarily convex) pseudo-circles.

This paper deals with the problem of reconstructing with guarantees an open subset Ω of Rd from an unorganized set S of sample points non-uniformly distributed inside Ω and not just lying on its boundary. We are able to select a subset of S making sense of what could be the boundary of a point cloud. The definition of this boundary is simple and there is an efficient algorithm to compute it. Under given conditions, we prove that this boundary carries topological and geometrical properties of Omega.