On $L_1$-embeddability of unions of $L_1$-embeddable metric spaces and of twisted unions of hypercubes

We prove that twisted unions of $L_1$-embedable metric spaces under some very general assumptions embed into $L_1$ with uniformly bounded distortions. In particular, the twisted unions of hypercubes studied by Johnson, Lindenstrauss, and Schechtman, and by Naor and Rabani embed into $L_1$ with bounded distortion. In the second part, we use the theory of stable metric spaces to find simple examples of metric spaces which are isometric to subsets of $L_p$ $(1\le p<\infty)$ but embeddings of their unions have distortion at least $3$.