On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson

Let $X$ be a normed space and $S_X$ be the unit sphere of $X$. We say that $X$ has Property (H) if there exist two increasing sequences $V_j\subset X$ and $H_j\subset \ell_2$, $j=1,2,\ldots$, of finite dimensional subspaces such that $V=\bigcup_j V_j$ is dense in $X$, and there is a uniformly contin