We study the quantum ( $$C^*$$ ) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, -extreme POVMs countable spaces (in particular for finite sets) are always spectral (normalized projection measures). More generally shown atomic spectral. A Krein–Milman type theorem ha...

1. Let F be a partially ordered vector space with an order unit e. It is well known that the class 5DÎ of maximal ideals of V is in one-one correspondence with the class K of normalized positive linear functionals, in the sense that to each ME^SR corresponds a positive linear functional <¡>m with M as its null-space and with M(e) = 1. A maximal ideal MG 90? is said to be extreme if <pM is an...

We give a short argument that for some C > 0, every n-dimensional Banach ball K admits a 256-round subquotient of dimension at least Cn/(logn). This is a weak version of Milman’s quotient of subspace theorem, which lacks the logarithmic factor. Let V be a finite-dimensional vector space over R and let V ∗ denote the dual vector space. A symmetric convex body or (Banach) ball is a compact convex...

It is proved that a discrete group G is amenable if and only if for every unitary representation of G in an infinite-dimensional Hilbert space H the maximal uniform compactification of the unit sphere SH has a G-fixed point, that is, the pair (SH, G) has the concentration property in the sense of Milman. Consequently, the maximal U(H)equivariant compactification of the sphere in a Hilbert space...

0. Introduction We work over an algebraically closed field k of characteristic 0. 0.1. Statement. In this paper, we use techniques of toric geometry to reprove the following theorem: Theorem 0.1. Let X be a projective variety of finite type over k, and let Z ⊂ X be a proper closed subset. Let G ⊂ Aut k (Z ⊂ X) be a finite group. Then there is a G-equivariant modification r : X 1 → X such that X...

Article history: Received 13 September 2016 Accepted 28 February 2017 Available online 6 March 2017 Communicated by E. Milman MSC: 35A15 35B40 35K55 47J10

Whitney's extension problem asks the following: Given a compact set E?Rn and function f:E?R, how can we tell whether there exists F?Cm(Rn) such that F=f on E? A 2006 theorem of Charles Fefferman [6] answers this question in its full generality. In paper, establish version adapted for variants Whitney problem, including nonnegative extensions smooth selection problems. Among other things, genera...

Motivated by typical questions from computational geometry (visibility and art gallery problems) and combinatorial geometry (illumination problems) we present an analogue of the Krein-Milman theorem for the class of star-shaped sets. If S ⊆ R is compact and star-shaped, we consider a fixed, nonempty, compact, and convex subset K of the convex kernel K0 = ck(S) of S, for instance K = K0 itself. ...