Vladimir P. Fonf

Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces

AbstractA closed, convex and bounded setP in a Banach spaceE is called a polytope if every finite-dimensional section ofP is a polytope. A Banach spaceE is called polyhedral ifE has an equivalent norm such that its unit ball is a polytope. We prove here:(1)LetW be an arbitrary closed, convex and bounded body in a separable polyhedral Banach spaceE and let ε>0. Then there exists a tangential ε-approximating polytopeP for the bodyW.(2)LetP be a polytope in a separable Banach spaceE. Then, for every ε>0,P can be ε-approximated by an analytic, closed, convex and bounded bodyV. We deduce from these two results that in a polyhedral Banach space (for instance in c0(ℕ) or inC(K) forK countable compact), every equivalent norm can be approximated by norms which are analytic onE/{0}.

Boundaries and generation of convex sets

AbstractWe introduce a notion which is intermediate between that of taking thew*-closed convex hull of a set and taking the norm closed convex hull of this set. This notion helps to streamline the proof (given in [FLP]) of the famous result of James in the separable case. More importantly, it leads to stronger results in the same direction. For example:1.AssumeX is separable and non-reflexive and its unit sphere is covered by a sequence of balls

Boundedly complete basic sequences,c 0-subspaces, and injections of Banach spaces

\left\{ {C_i } \right\}_{i = 1}^\infty

Covering a Banach space

of radiusa<1. Then for every sequence of positive numbers

Coverings of Banach spaces: beyond the Corson theorem

\left\{ {\varepsilon _i } \right\}_{i = 1}^\infty

Corrigendum to “Best approximation in polyhedral Banach spaces” [J. Approx. Theory 163 (11) (2011) 1748–1771]

tending to 0 there is anf εX*, such that ‖f‖ = 1 andf (x)≤1 −εi, wheneverx εCi,i=1,2,…2.AssumeX is separable and non-reflexive and letT:Y →X* be a bounded linear non-surjective operator. Then there is anf εX* which does not attain its norm onBX such thatf ∉T(Y).