Extensions and analogues of the Brunn–Minkowski theory
The Brunn–Minkowski theory, developed in previous chapters, is based on the Minkowski addition of convex sets, combined with the notion of volume. One can replace Minkowski addition by other combinations of convex bodies or star bodies, and in this way generalize or dualize certain parts of the classical theory. The basic facts of such extensions and analogues will be described in the first four sections of this chapter. We point out that, in these variations of the Brunn–Minkowski theory, the position of the origin of the space plays an essential role; the classical theory remains the only one that is translation invariant.
The fifth section describes first steps on a different route of ...