It is shown that if a maximal operator associated with a homothecy invariant collection of convex sets Rn satisfies Cordoba-Fefferman Tauberian condition at some fixed level, then it must satisfy the same condition at all levels and moreover the maximal operator is Lp − bounded for sufficiently large p. As a corollary of these results it is shown that any density basis that is a homothecy invariant collection of convex sets in Rn must differentiate integrals of the functions from Lp for sufficiently large p. This is a joint result with Paul Hagelstein.