Lipschitz Free Spaces and Subsets of Finite-Dimensional Spaces

We consider two questions on the geometry of Lipschitz free $p$-spaces $\mathcal F_p$, where $0<p\leq 1$, over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal M, \rho)$ is an infinite doubling metric space (e.g., an infinite subset of an Euclidean space), then $\mathcal F_p (\mathcal M, \rho^\alpha)\simeq\ell_p$ for every $\alpha\in(0,1)$ and $0<p\leq 1$. An upper bound on the Banach-Mazur distance between the spaces $\mathcal F_p ([0, 1]^d, |\cdot|^\alpha)$ and $\ell_p$ is given. Moreover, we tackle a question due to arXiv:2006.08018v1 [math.FA] and expound the role of $p$, $d$ for the Lipschitz constant of a canonical, locally coordinatewise affine retraction from $(K, |\cdot|_1)$, where $K=\bigcup_{Q\in \mathcal R} Q$ is a union of a collection $\emptyset \neq \mathcal R \subseteq \{ Rw + R[0,1]^d: w\in\mathbb Z^d\}$ of cubes in $\mathbb R^d$ with side length $R>0$, into the Lipschitz free $p$-space $\mathcal F_p (V, |\cdot|_1)$ over their vertices.