Stable unit balls in Orlicz spaces

Abstract:

Let ${L^\phi }(\mu )$ be an Orlicz space and $X \subseteq {L^\phi }(\mu )$ an ideal such that ${I_\phi }(f/||f||) = 1$ for each $f \in X\backslash \left \{ 0 \right \}$. Then the unit ball ${B_X}$ is stable, that is, the midpoint map ${\Phi _{1/2}}:{B_X} \times {B_X} \to {B_X}$ defined by ${\Phi _{1/2}}(x,y) = \tfrac {1}{2}(x + y)$, is open. In particular, ${B_E}\phi$ is stable, ${E^\phi }$ being the subspace of finite elements of ${L^\phi }(\mu )$ (i.e., $f \in {E^\phi }$ iff ${I_\phi }(\lambda f) < + \infty$ for each $\lambda > 0$), and ${B_{{L^\phi }(\mu )}}$ is stable when $\phi$ satisfies condition $({\Delta _2})$ or $({\delta _2})$, depending on the measure $\mu$.