Renormings of $c_0$ and the minimal displacement problem

Abstract

The aim of this paper is to show that for every Banach space \((X, \|\cdot\|)\) containing asymptotically isometric copy of the space \(c_0\) there is a bounded, closed and convex set \(C \subset X\) with the Chebyshev radius \(r(C) = 1\) such that for every \(k \geq 1 \) there exists a \(k\)-contractive mapping \(T : C \to C\) with \(\| x - Tx \| > 1 − 1/k\) for any \(x \in C\).