  In this paper, we first introduce a new concept of {\it dual quermassintegral
sum function} of two star bodies and establish Minkowski's type inequality for
dual quermassintegral sum of mixed intersection bodies, which is a general form
of the Minkowski inequality for mixed intersection bodies. Then, we give the
Aleksandrov--Fenchel inequality and the Brunn--Minkowski inequality for mixed
intersection bodies and some related results. Our results present, for
intersection bodies, all dual inequalities for Lutwak's mixed prosection bodies
inequalities.
