Siano , sottoinsiemi convessi, chiusi e limitati di uno spazio normato , con le frontiere , . Dimostriamo che , dove è la metrica di Hausdorff tra sottoinsiemi chiusi di . Studiamo inoltre la continuità e la semicontinuità superiore ed inferiore di una multifunzione di tipo «frontiera».
On définit un filtre coanalytique sur les entiers tel que pour tout convexe compact métrisable , les fonctions boréliennes fortement affine sur , i.e. vérifiant les égalités barycentriques, soient exactement les limites simples suivant ce filtre de suites de fonctions affines continues sur .
Characterizations of extreme infinite symmetric stochastic matrices with respect to arbitrary non-negative vector r are given.
Let , and be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of and the space are AR. In case X is separable, is locally path-connected.
Let X be a compact convex set and let ext X stand for the set of all extreme points of X. We characterize those bounded function defined on ext X which can be extended to an affine Baire-one function on the whole set X.
We show that the space of tracial quantum symmetric states of an arbitrary unital C*-algebra is a Choquet simplex and is a face of the tracial state space of the universal unital C*-algebra free product of A with itself infinitely many times. We also show that the extreme points of this simplex are dense, making it the Poulsen simplex when A is separable and nontrivial. In the course of the proof we characterize the centers of certain tracial amalgamated free product C*-algebras.
Sufficient conditions are given in order that, for a bounded closed convex subset of a locally convex space , the set of continuous functions from the compact space into , is the uniformly closed convex hull in of its extreme points. Applications are made to the unit ball of bounded (or compact, or weakly compact) operators from certain Banach spaces into .
We prove that each non-separable completely metrizable convex subset of a Fréchet space is homeomorphic to a Hilbert space. This resolves a more than 30 years old problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowolski and Toruńczyk, this result implies that each closed convex subset of a Fréchet space is homeomorphic to for some cardinals 0 ≤ n ≤ ω, 0 ≤ m ≤ 1 and κ ≥ 0.