On coincidence of Pettis and McShane integrability

summary:R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$, every Pettis integrable function $f\colon [0,1]\rightarrow X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0,1]$ into $X$, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0,1]$ (mostly) into $C(K)$ spaces. We focus in more detail on the behavior of several concrete Eberlein (Corson) compact spaces $K$, that are not uniform Eberlein, with respect to the integrability of some natural scalarly negligible functions from $[0,1]$ into $C(K)$ in McShane sense

Reznichenko families of trees and their applications

AbstractExamples of Talagrand, Gul'ko and Corson compacta resulting from Reznichenko families of trees are presented. The Kσδ property for weakly K-analytic Banach spaces with an unconditional basis is proved

Talagrand's Kσ δ problem

AbstractWe show that the Banach spaces C(K) with K either an adequate Talagrand compact or a quasi adequate σ-Eberlein Talagrand compact are Kσδ subsets of their second dual endowed with the weak∗ topology. As consequence we obtain that weakly K-analytic Banach spaces with an unconditional basis are Kσδ. We also provide an example of a Talagrand compact K such that C(K) is not Kσδ in its second dual. This answers a problem posed by M. Talagrand

Combinatorial distance geometry in normed spaces

We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as minimum spanning trees and Steiner minimum trees. In particular, we discuss translative kissing (or Hadwiger) numbers, equilateral sets, and the Borsuk problem in normed spaces. We show how to use the angular measure of Peter Brass to prove various statements about Hadwiger and blocking numbers of convex bodies in the plane, including some new results. We also include some new results on thin cones and their application to distinct distances and other combinatorial problems for normed spaces