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The aim of this paper is to show that for every Banach space (X, || · ||) containing asymptotically isometric copy of the space c0 there is a bounded, closed and convex set C ⊂ X with the Chebyshev radius r(C) = 1 such that for every k ≥ 1 there exists a k-contractive mapping T : C → C with [...] for any x ∊ C.

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

For every predual $X$ of ${\ell }_1$ such that the standard basis in ${\ell }_1$ is weak${}^{*}$ convergent, we give explicit models of all Banach spaces $Y$ for which the Banach-Mazur distance $d(X,Y)=1$. As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space ${\ell }_1$, with a predual $X$ as above, has the stable weak${}^{*}$ fixed point property if and only if it has almost stable weak${}^{*}$ fixed point property, i.e. the dual $Y^{*}$ of every Banach space $Y$ has the weak${}^{*}$ fixed point property...