The weak stability of the positive face in $L^ 1$

Abstract:

Let F be the positive face of the unit ball of ${L^1}[0,1]$. We show that F is weakly stable in the sense that the midpoint map ${\Phi _{1/2}}:F \times F \to F$, with ${\Phi _{1/2}}(f,g) = \frac {1}{2}(f + g)$, is open with respect to the weak topology. This weak stability of the set F is the reason behind the fact that the notions of "huskable" and "strongly regular" operators coincide for operators from ${L^1}[0,1]$ to a Banach space X. We prove this stability by showing that if ${f_1},{f_2} \in F,\lambda \in (0,1),\varepsilon > 0$ and $\delta \geq \max \{ 2\varepsilon /\lambda ,2\varepsilon /(1 - \lambda )\}$, then \[ \lambda {V_{P,\delta }}({f_1}) + (1 - \lambda ){V_{P,\delta }}({f_2}) \supset {V_{P,\varepsilon }}[\lambda {f_1} + (1 - \lambda ){f_2}],\] where $P = \{ {A_1}, \ldots ,{A_n}\}$ is a finite positive partition of [0, 1] and \[ {V_{P,\varepsilon }}(f) = \left \{ {g \in F:\sum \limits _{i = 1}^n {\left | {\int _{{A_i}} {(f - g)(t)d\mu (t)} } \right | \leq \varepsilon } } \right \}\] for any f in F. We construct an example showing that for any $0 < \lambda < 1$ there are functions ${f_1}$ and ${f_2}$ in F such that if $0 < \varepsilon < 2\min \{ \lambda ,1 - \lambda \}$ and $0 \leq \delta < \max \{ \varepsilon /\lambda ,\varepsilon /(1 - \lambda )\}$, then \[ \lambda {V_{P,\delta }}({f_1}) + (1 - \lambda ){V_{P,\delta }}({f_2})\not \supset {V_{P,\varepsilon }}(\lambda {f_1} + (1 - \lambda ){f_2}).\] Thus the "formula" that $\lambda {V_{p,\varepsilon }}({f_1}) + (1 - \lambda ){V_{p,\varepsilon }}({f_2}) = {V_{p,\varepsilon }}(\lambda {f_1} + (1 - \lambda ){f_2})$ given by Ghoussoub et al. in Mem. Amer. Math. Soc., vol. 70, no. 378, which is used there to establish the weak stability of F, is false.