Under a suitable renorming every nonreflexive Banach space has a finite subset without a Steiner point
Vladimir M. Kadets · Matematychni Studii · 2011
V. Kadets.Under a suitable renorming every nonreflexive Banach space has a finite subset without a Steiner point, Mat.Stud.36 (2011), 197-200.We present a refinement of the recent Borodin's example of a finite set without a Steiner point.Namely, we show that under a suitable renorming such an example exists in every nonreflexive Banach space.В. Кадец.Каждое нерефлексивное банахово пространство в подходящей перенормировке содержит конечное множество без точек Штейнера // Мат.Студiї.-2011.-Т.36, №2.-C.197-200.Недавно П.А.Бородин построил пример конечного множества в банаховом пространстве, не имеющего точек Штейнера.Мы уточняем этот результат, показывая, что в подходящей эквивалентной перенормировке такие примеры есть в любом нерефлексивном банаховом пространстве.For any finite collection A = {x 1 , . . ., x n } of (not necessarily distinct) elements of a Banach space X a Steiner point of A is every point s ∈ X at which the function x → ∑ n k=1 ∥x -x k ∥ attains its minimum.Let us say that a Banach space X has the Steiner Point Property (X ∈ StPP) if every finite collection A ⊂ X possesses a Steiner point.By weak compactness argument every reflexive space has the StPP (see [1] for the corresponding references and for a short proof).The class of spaces with the Steiner Point Property contains also some non-reflexive spaces, like dual spaces, L 1 [0, 1], or more generally every Banach space that is 1-complemented in its bidual (see Theorem 1 below).The problem whether C[0, 1] in its original norm has the StPP remains open.Recently, P. A. Borodin [1] presented the first example of a Banach space X that does not enjoy the StPP.This example is obtained by introducing an equivalent norm on C[0, 1] that "mixes" in a clever way the original norm of C[0, 1] with the L 1 -norm.In this short note we use the idea of Borodin's construction in order to show that in every nonreflexive Banach space X there is an equivalent norm ∥ • ∥ b such that (X, ∥ • ∥ b ) / ∈ StPP.In the sequel, if X is a Banach space then B X stands for its closed unit ball, X * and X * * stand for the dual and bidual spaces respectively.The norm closure of a subset D ⊂ X we denote cl(D).We use the word "operator" for bounded linear operators.A Banach space X is said to be 1-complemented in its bidual if there is a linear projection P : X * * → X with ∥P ∥ = 1.For standard facts about Banach spaces and properties of weak and weak* topologies we refer to [2], for more advanced Banach space theory results we refer to [3].We start with a simple positive result.