Points minimaux et ensembles optimaux dans les espaces de Banach

Almost constrained subspaces of Banach spaces

“…Our condition is in terms of functionals with "locally unique" Hahn-Banach (i.e., norm-preserving) extensions, which improves significantly upon all existing conditions of similar type, as noted in [3,8], and has significantly simpler proof. As in [6,7], these conditions actually imply the existence of a unique norm 1 projection.…”

Sets of minimal points in 𝑙_{𝑝}

“…By the argument used in [1], we have the inclusions sat{B¡ ) ç S and sat(7J/(/v)) Ç5n/j , so it will suffice to prove the reverse inclusion.…”

“…The set of all points minimal with respect to M is called the minimal hull of M and is denoted by min(M). The minimal hull is somehow a generalization of the closed convex hull, since in Hubert spaces they are the same [1].…”

“…Let X = {mx, m2, xx, ... , xN, ...} be any countable set. We provide X with a metric d as follows: (1) d{mi,rt}2) = 2, (2) d(xL,xt) = l íi.éj), It is easily seen that this actually defines a metric on X (we only have to worry about the triangle inequality). Now look at the compact set M = {mx , m2}.…”

Contractive and optimal sets in Musielak-Orlicz spaces with a smoothness condition

“…The latter is closely related to the notion of optimal sets (Definition 5.2) introduced by P. Enflo (cf. [8,9] and [7]) and developed by B. Beauzamy in [2]. This is used to study the approximation in norm in Banach spaces.…”