We consider the expected number of Voronoi vertices (or number of Delaunay cells for the dual structure) 
for a set of n i.i.d. random point sites chosen uniformly from the unit d-hypercube. We show an upper
bound for this number which is linear in n, the number of random point sites, where d  is assumed to
be a constant. This result matches the trivial lower bound of n. This is an open problem since several
years. In 1991, Dwyer showed that for a uniform distribution from the unit d-ball the average number
of Voronoi vertices is linear in n and it is commonly assumed that this holds for any reasonable
probability distribution.