  We present improved upper bounds for the size of relative
(p,Epsilon)-approximation for range spaces with the following property: For any
(finite) range space projected onto (that is, restricted to) a ground set of
size n and for any parameter 1 <= k <= n, the number of ranges of size at most
k is only nearly-linear in n and polynomial in k. Such range spaces are called
"well behaved". Our bound is an improvement over the bound O(\log{(1/p)/\eps^2
p) introduced by Li etal. for the general case (where this bound has been shown
to be tight in the worst case), when p << Epsilon. We also show that such small
size relative (p,Epsilon)-approximations can be constructed in expected
polynomial time.
  Our bound also has an interesting interpretation in the context of "p-nets":
As observed by Har-Peled and Sharir, p-nets are special cases of relative
(p,Epsilon)-approximations. Specifically, when Epsilon is a constant smaller
than 1, their analysis implies that there are p-nets of size O(\log{(1/p)}/p)
that are \emph{also} relative approximations. In this context our construction
significantly improves this bound for well-behaved range spaces. Despite the
progress in the theory of p-nets and the existence of improved bounds
corresponding to the cases that we study, these bounds do not necessarily
guarantee a bounded relative error.
  Lastly, we present several geometric scenarios of well-behaved range spaces,
and show the resulting bound for each of these cases obtained as a consequence
of our analysis. In particular, when Epsilon is a constant smaller than 1, our
bound for points and axis-parallel boxes in two and three dimensions, as well
as points and "fat" triangles in the plane, matches the optimal bound for
p-nets.
