Renormings of c 0 and the minimal displacement problem

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 ‖x− Tx‖ > 1− 1 k for any x ∈ C.


Introduction and Preliminaries.
Let C be a nonempty, bounded, closed and convex subset of an infinitely dimensional real Banach space (X, • ).The Chebyshev radius of C relative to itself is the number We say that a mapping T : C → C satisfies the Lipschitz condition with a constant k or is k-lipschitzian, if for all x, y ∈ C, T x − T y ≤ k x − y .
The smallest constant k for which the above inequality holds is called the Lipschitz constant for T and it is denoted by k(T ).By L(k) we denote the class of all k-lipschitzian mappings T : C → C. A mapping T : C → C is called k-contractive if for all x, y ∈ C, x = y, we have The minimal displacement problem has been raised by Goebel in 1973, see [6].Standard situation is the following.For any k-lipschitzian mapping T : C → C the minimal displacement of T is the number given by It is known that for every k ≥ 1 (for the proof see for example [8]) For any set C we define the function Consequently, for every k ≥ 1, The function ϕ C is called the characteristic of minimal displacement of C.
If C is the closed unit ball B X , then we write ψ X instead of ϕ B X .We also define the characteristic of minimal displacement of the whole space X as Hence, for every k > 1, The minimal displacement problem is the task to find or evaluate functions ϕ and ψ for concrete sets or spaces.Obviously this problem is matterless in the case of compact set C because, in view of the celebrated Schauder's fixed point theorem, we have ϕ C (k) = 0 for any k > 1.If C is noncompact then by the theorem of Sternfeld and Lin [12] we get ϕ C (k) > 0 for all k > 1. Hence we additionally assume that C is noncompact and we restrict our attention to the class of lipschitzian mappings with k(T ) ≥ 1.
The set C for which ϕ C (k) = 1 − 1 k r(C) for every k > 1 is called extremal (with respect to the minimal displacement problem).There are examples of spaces having extremal balls.Among them are spaces of continuous functions C[a, b], bounded continuous functions BC(R), sequences converging to zero c 0 , all of them endowed with the standard uniform norm (see [7]).Recently the present author [13] proved that also the space c of converging sequences with the sup norm has extremal balls.It is still unknown if the space l ∞ of all bounded sequences with the sup norm has extremal balls.Very recently Bolibok [1] proved that The same estimate holds for the space of summable functions (equivalent classes) L 1 (0, 1) equipped with the standard norm (see [2]) as well as few other spaces (see [9]).
In the case of space l 1 of all summable sequences with the classical norm we have: Nevertheless, the subset for every k > 1 (for the proof see [7]).
In this paper we deal with a problem of existence of extremal sets in spaces containing isomorphic copies of c 0 .Obviously, for every such space X we have ϕ X (k) = 1 − 1 k as an immediate consequence of the following theorem by James [10], its stronger version states: Theorem 1.1 (James's Distortion Theorem, stronger version).A Banach space X contains an isomorphic copy of c 0 if and only if, for every null sequence { n } ∞ n=1 in (0, 1), there exists a sequence {x n } ∞ n=1 in X such that holds for all {t n } ∞ n=1 ∈ c 0 and for all k = 1, 2, . . . .However, it is not known if all isomorphic copies of c 0 contain an extremal subset.We shall prove that the answer is affirmative in the case of spaces containing an asymptotically isometric copies of c 0 .This class of spaces has been introduced and widely studied by Dowling, Lennard and Turett (see [11], Chapter 9).Let us recall that a Banach space X is said to contain an asymptotically isometric copy of c 0 if for every null sequence for all {t n } ∞ n=1 ∈ c 0 .Dowling, Lennard and Turett proved the following theorems.Theorem 1.2 (see [3] or [11]).If a Banach space X contains an asymptotically isometric copy of c 0 , then X fails the fixed point property for nonexpansive (and even contractive) mappings on bounded, closed and convex subsets of X. Theorem 1.3 (see [5] or [11]).If Y is a closed infinite dimensional subspace of (c 0 , • ∞ ), then Y contains an asymptotically isometric copy of c 0 .Theorem 1.4 (see [5]).Let Γ be an uncountable set.Then every renorming of c 0 (Γ) contains an asymptotically isometric copy of c 0 .
Let us recall that a mapping T : C → C is said to be asymptotically nonexpansive if T n x − T n y ≤ k n x − y for all x, y ∈ C and for all n = 1, 2, . . ., where {k n } ∞ n=1 is a sequence of real numbers with lim n→∞ k n = 1.
Now we are ready to cite the following theorem.
Theorem 1.5 (see [4] or [11]).If a Banach space X contains an isomorphic copy of c 0 , then there exists a bounded, closed, convex subset C of X and an asymptotically nonexpansive mapping T : C → C without a fixed point.
In particular, c 0 cannot be renormed to have the fixed point property for asymptotically nonexpansive mappings.
Remark 1.6 (see [5] or [11]).There is an isomorphic copy of c 0 which does not contain any asymptotically isometric copy of c 0 .

Main result.
Theorem 2.1.If a Banach space X contains an asymptotically isometric copy of c 0 , then there exists a bounded, closed and convex subset C of X with r(C) = 1 such that for every k ≥ 1 there exists a k-contractive mapping It is clear that C is a bounded, closed and convex subset of X.
We claim that r(C) = 1.Fix w = ∞ i=1 t i y i ∈ C and let {z n } ∞ n=2 be a sequence of elements in C defined by Letting n → ∞, we get r(w, C) := sup{ w − x : x ∈ C} ≥ 1 for any w ∈ C and consequently r(C) ≥ 1.
Now let {z n } ∞ n=1 be a sequence in C given by Then for every w = ∞ i=1 t i y i ∈ C we have To construct desired mapping T we shall need the function α : [0, ∞) → [0, 1] defined by It is clear that the function α satisfies the Lipschitz condition with the constant 1, that is, for all s, t ∈ [0, ∞) we have For arbitrary k ≥ 1 we define a mapping T : Then for any w = ∞ i=1 t i y i and z = ∞ i=1 s i y i in C such that w = z we have Hence the mapping T is k-contractive.
We claim that for every Indeed, suppose that there exists w = ∞ i=1 t i y i ∈ C such that w − T w ≤ 1 − 1 k , that is, This implies that Hence t i ≥ 1 k for i = 1, 2, . . . .But {t i } ∞ i=1 ∈ c 0 , a contradiction.
Corollary 2.2.If a Banach space X contains an asymptotically isometric copy of c 0 , then X contains an extremal subset.
Corollary 2.3.If Y is a closed infinite dimensional subspace of (c 0 , • ∞ ), then Y contains an extremal subset.