Isometric embeddings of Banach spaces under optimal projection constants

Let $X$ be a Banach space with separable dual. It is proved that for every $\varepsilon\in (0,1)$, $X$ embeds isometrically into a Banach space $W$ with a shrinking basis $(w_n)$ which is $(1+ \varepsilon)$-monotone. Moreover, if $X$ has further an FDD $(E_n)$ whose strong bimonotonicity projection constant is not larger than $\mathcal{D}$, then $(w_n)$ has strong bimonotonicity projection constant not exceeding $\mathcal{D}(1 +\varepsilon)$. Further, if $(E_n)$ is $\mathcal{C}$-unconditional then $(w_n)$ is $\mathcal{C}(1 + \varepsilon)$-unconditional. The proof uses renorming and skipped blocking decomposition techniques. As an application, we prove that every Banach space having a shrinking $\mathcal{D}$-unconditional basis with $\mathcal{D}<\sqrt{6}-1$, has the weak fixed point property.

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