New matrix inequalities for Firey's extension of Minkowski and Brunn-Minkowski inequalities

The Brunn-Minkowski theory is a central part of convex geometry. At its foundation lies the Minkowski addition of convex bodies which led to the definition of mixed volume of convex bodies and to various notions and inequalities in convex geometry. Various matrix analogs of these notions and inequalities have been well known for over a century. We present a few new analogs. The major theorems presented here are the matrix analogs of Firey's Extension of Minkowski inequality and Firey's Extension of Brunn-Minkowski inequality.