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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 vectorspaces. We solve an open problem and show that if $(\mathcal M, \rho)$ is aninfinite doubling metric space (e.g., an infinite subset of an Euclideanspace), 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 distancebetween the spaces $\mathcal F_p ([0, 1]^d, |\cdot|^\alpha)$ and $\ell_p$ isgiven. Moreover, we tackle a question due to arXiv:2006.08018v1 [math.FA] andexpound the role of $p$, $d$ for the Lipschitz constant of a canonical, locallycoordinatewise 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 $\mathcal F_p (V, |\cdot|_1)$ overtheir vertices.

Lipschitz Free Spaces and Subsets of Finite-Dimensional Spaces