Let P(X,Y) be any fixed probability density in the plane that is unimodal on lines. For any such density I conjecture these upper bounds: (i) sampling N points from it independently will yield expected #vertices on convex hull growing like O(N^(1/3)) and (ii) expected #undominated points growing like O(N^(1/2)). A point (X,Y) is "undominated" if the infinite closed axis-parallel rectangle with lower left corner (X,Y) is empty of other points. These bounds (if valid) are tight since the uniform density within a circle (zero outside) achieves them. The variances might (more strongly) be conjectured also to be bounded by the same upper bounds. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)