Stable and uniformly stable unit balls in Banach spaces

Abstract

Let X X be a Banach space with closed unit ball B X {B_X} and, for x ∈ X x \in X , r ≥ 0 r \geq 0 , put B ( x ; r ) = { u ∈ X : | | u − x | | ≤ r } B(x;r)= \{ u \in X:||u - x|| \leq r\} and V ( x , r ) = B X ∩ B ( x ; r ) V(x,r)= {B_X} \cap B(x;r) . We say that B X {B_X} (or in general a convex set) is stable if the midpoint map Φ 1 / 2 : B X × B X → B X {\Phi _{1/2}}:{B_X} \times {B_X} \to {B_X} , with Φ 1 / 2 ( u , υ ) = 1 2 ( u + υ ) {\Phi _{1/2}}(u,\upsilon )= \frac {1}{2}(u + \upsilon ) , is open. We say that B X {B_X} is uniformly stable (US) if there is a map α : ( 0 , 2 ] → ( 0 , 2 ] \alpha :(0,2] \to (0,2] , called a modulus of uniform stability, such that, for each x , y ∈ B X x,y \in {B_X} and r ∈ ( 0 , 2 ] , V ( 1 2 ( x + y ) ; α ( r ) ) ⊆ 1 2 ( V ( x ; r ) + V ( y ; r ) ) r \in (0,2],V(\frac {1} {2}(x + y);\alpha (r)) \subseteq \frac {1} {2}(V(x;r) + V(y;r)) . Among other things, we see: (i) if dim ⁡ X ≥ 3 \dim X \geq 3 , then X X admits an equivalent norm such that B X {B_X} is not stable; (ii) if dim ⁡ X > ∞ \dim X > \infty , B X {B_X} is stable iff B x {B_x} is US; (iii) if X X is rotund, X X is uniformly rotund iff B X {B_X} is US; (iv) if X X is 3.2. I.P 3.2.{\text {I.P}} , B X {B_X} is US and α ( r ) = r / 2 \alpha (r)= r/2 is a modulus of US; (v) B X {B_X} is US iff B X ∗ ∗ {B_{{X^{ \ast \ast }}}} is US and X X , X ∗ ∗ {X^{ \ast \ast }} have (almost) the same modulus of US; (vi) B X {B_X} is stable (resp. US) iff B C ( K , X ) {B_{C(K,X)}} is stable (resp. US) for each compact K K iff B A ( K , X ) {B_{A(K,X)}} is stable (resp. US) for each Choquet simplex K K ; (vii) B X {B_X} is stable iff B L p ( μ , X ) {B_{{L_p}(\mu ,X)}} is stable for each measure μ \mu and 1 ≤ p > ∞ 1 \leq p > \infty .