On Property (H) of $c_0$ and the Sphere 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 $V_j\subset X$,;$H_j\subset {\mathrm{\ell }}_2,j=1,2,\cdots$ of finite dimensional subspaces satisfying the union $V\equiv {\bigcup }_j{V}_j$ is dense in $X$ and there is a uniform continuous mapping $\psi :S_V\to S_{{\mathrm{\ell }}_2}$ so that the restrictions $\psi {\mathrm{\mid }}_{S_{V_j}},j=1,2,\cdots$ are isomorphisms from $S_{V_j}$ to $S_{H_j}$. The Kasparov-Yu problem is whether the Banach space $c_0$ of all null sequences admits Property (H), or weaker, rational Property (H). In this paper, we show that $c_0$ does not have rational Property (H). Meanwhile, we show the following results, which solve a problem of Gromov and Johnson.