We give a useful new characterization of the set of all completely positive, trace-preserving maps Φ : M 2 → M 2 from which one can easily check any trace-preserving map for complete positivity. We also determine explicitly all extreme points of this set, and give a useful parameterization after reduction to a certain canonical form. This allows a detailed examination of an important class of non-unital extreme points that can be characterized as having exactly two images on the Bloch sphere.We also discuss a number of related issues about the images and the geometry of the set of stochastic maps, and show that any stochastic map on M 2 can be written as a convex combination of two "generalized" extreme points.

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The surface body is a generalization of the floating body. Its relation to p-affine surface area is studied.