Deville's Master Lemma and Stone's Discreteness in Renorming Theory

Banach spaces $X$ with an equivalent $\sigma(X,F)$-lower semicontinuous and locally uniformly rotund norm, for a norming subspace $F\subset X^*$, are those spaces $X$ that admit countably many families of convex and $\sigma(X,F)$-lower semicontinuous functions $\{\varphi_i^n:X \rightarrow {\mathbb R}^+; i \in I_n\}_{n=1}^\infty$ such that there are open subsets $$G_i^n \subset \{\varphi_i^n >0\} \cap\{\varphi_j^n =0: j\neq i, j \in I_n\}$$ with $\{G_i^n: i\in I_n, n\in {\mathbb N}\}$ a basis for the norm topology of $X$.