Elliptic Brunn-Minkowski theory: Duality and applications

The Brunn-Minkowski theory is the core of convex geometry. In contrast to convex bodies studied by the Brunn-Minkowski theory, elliptic Brunn-Minkowski theory employs matrices as its basic research object.;It is the aim of this work to develop the Brunn-Minkowski theory for positive definite symmetric matrices, which is analogous to the classical Brunn-Minkowski theory in convex geometry and to establish a dual theory for the elliptic Brunn-Minkowski theory. Some applications and related topics are also studied.