Elisabeth Werner

An analysis of completely-positive trace-preserving maps on M2

Abstract 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.

New Lp affine isoperimetric inequalities

We prove new Lp affine isoperimetric inequalities for all p[-8,1). We establish, for all p?-n, a duality formula which shows that Lp affine surface area of a convex body K equals affine surface area of the polar body K?.

Relative entropy of cone measures and Lp centroid bodies

Let K be a convex body in R n . We introduce a new affine invariant, which we call ΩK ,t hat can be found in three different ways: (a) as a limit of normalized Lp-affine surface areas; (b) as the relative entropy of the cone measure of K and the cone measure of K ◦ ; (c) as the limit of the volume difference of K and Lp-centroid bodies. We investigate properties of ΩK and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show an ‘information inequality’ for convex bodies.

New

We prove new Lp affine isoperimetric inequalities for all p[-8,1). We establish, for all p?-n, a duality formula which shows that Lp affine surface area of a convex body K equals affine surface area of the polar body K?.

L_p

The goal of this note is to show that Hastings’s counterexample to the additivity of minimal output von Neumann entropy can be readily deduced from a sharp version of Dvoretzky’s theorem.

Affine Isoperimetric Inequalities

Motivated by the Blaschke-Santalo inequality, we define for a convex body K in R and for t ∈ R the Santalo-regions S(K,t) of K. We investigate properties of these sets and relate them to a concept of Affine Differential Geometry, the affine surface area of K. Let K be a convex body in R. For x ∈ int(K), the interior of K, let K be the polar body of K with respect to x. It is well known that there exists a unique x0 ∈ int(K) such that the product of the volumes |K||K0 | is minimal (see for instance [Sch]). This unique x0 is called the Santalo-point of K. Moreover the Blaschke-Santalo inequality says that |K||K0 | ≤ v n (where vn denotes the volume of the n-dimensional Euclidean unit ball B(0, 1)) with equality if and only if K is an ellipsoid. For t ∈ R we consider here the sets S(K, t) = {x ∈ K : |K||K | v2 n ≤ t}. Following E. Lutwak, we call S(K, t) a Santalo-region of K. Observe that it follows from the Blaschke-Santalo inequality that the Santalopoint x0 ∈ S(K, 1) and that S(K, 1) = {x0} if and only if K is an ellipsoid. Thus S(K, t) has non-empty interior for some t < 1 if and only if K is not an ellipsoid. In the first part of this paper we show some properties of S(K, t) and give estimates on the “size” of S(K, t). This question was asked by E. Lutwak. ∗the paper was written while both authors stayed at MSRI †supported by a grant from the National Science Foundation. MSC classification 52

Hastings’s Additivity Counterexample via Dvoretzky’s Theorem

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed p-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of Lp affine surface areas, mixed p-affine surface areas and other functionals via these bodies. The surprising new element is that not necessarily convex bodies provide the tool for these interpretations.

The Santaló-regions of a convex body

Let K be a convex body in \(\mathbb{R}^n\) and let \(f : \partial K \to \mathbb{R}_ + \) be a continuous, positive function with \(\int_{\partial K} f(x )d\mu_{\partial K} (x ) = 1\) where \(\mu_{\partial K}\) is the surface measure on \(\partial K\). Let \(\mathbb{P}_f\) be the probability measure on \(\partial K\) given by \({\rm d}\mathbb{P}_f(x ) = f (x ){\rm d} \mu_{\partial K} (x )\). Let \(\kappa\) be the (generalized) Gaus-Kronecker curvature and \(\mathbb{E}(f,N )\) the expected volume of the convex hull of N points chosen randomly on \(\partial K\) with respect to \(\mathbb{P}_f\). Then, under some regularity conditions on the boundary of K

Polytopes with Vertices Chosen Randomly from the Boundary of a Convex Body

\lim_{ N\to \infty} \frac{{\rm vol}_n (K ) -\mathbb{E} (f,N )}{\left(\frac{1}{N}\right)^{\frac{2}{n-1}}} = c_n\int_{\partial K}\frac{\kappa(x)^{\frac{1}{n-1}}}{f(x)^{\frac{2}{n-1}}} {\rm d}\mu_{\partial K}(x),