Erdös-Szekeres "happy end"-type theorems for separoïds

In 1935 Pal Erdos and Gyorgy Szekeres proved that, roughly speaking, any configuration ofnpoints in general position in the plane havelognpoints in convex position - which are the vertices of a convex polygon. Later, in 1983, Bernhard Korte and Laszlo Lovasz generalised this result in a purely combinatorial context; the context of greedoids. In this note we give one step further to generalise this last result for arbitrary dimensions, but in the context of separoids; thus, via the geometric representation theorem for separoids, this can be applied to families of convex bodies. Also, it is observed that the existence of some homomorphisms of separoids implies the existence of not-too-small polytopal subfamilies - where each body is separated from its relative complement. Finally, by means of a probabilistic argument, it is settled, basically, that for all d>2, asymptotically almost all ''simple'' families of n ''d-separated'' convex bodies contains a polytopal subfamily of order lognd+1.