  Generalizing techniques that prove that Veronese subrings are Koszul, we show
that Rees and multi-Rees algebras of certain types of principal strongly stable
ideals are Koszul. We provide explicit Gr\"obner basis for the defining ideals
of these multi-Rees algebras with squarefree initial monomials, to show that
they are also normal Cohen-Macaulay domains.
