On the strongly subdifferentiable points in Lipschitz-free spaces

In this paper, we present some sufficient conditions on a metric space $M$ for which every molecule is a strongly subdifferentiable (SSD, for short) point in the Lipschitz-free space $\mathcal{F}(M)$ over $M$. Our main result reads as follows: if $(M,d)$ is a metric space and $\gamma >0$, then there exists a (not necessarily equivalent) metric $d_{\gamma }$ in $M$ such that every finitely supported element in $\mathcal{F}(M,d_{\gamma })$ is an SSD point. As an application of the main result, it follows that if $M$ is uniformly discrete and $\epsilon >0$ is given, there exists a metric space $N$ and a $(1+\epsilon )$-bi-Lipschitz map $\phi :M\to N$ such that the set of all SSD points in $\mathcal{F}(N)$ is dense.