Stable and uniformly stable unit balls in Banach spaces

Abstract:

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