Q: p-norm and relative relations questions Let $\left | x \right |_{p}=(|x_1|^p+...+|x_n|^p)^{1/p}$ denote the $p$-norm of the vector $x\in\mathbb{R}_n$, $\infty\ge p\ge1$. 
Let $B_{r}^{p}$={$x\in\mathbb{R}_n : |x|_p\leq r$}.
1)Is there a $c>0$ such that $B_{1/c}^{2}\subseteq B_{1}^{\infty }\subseteq B_{c}^{2}$? If so, what is the smallest such $c$?
2)Given $1≤p<q<∞$, is there a $c>0$ such that $B_{1/c}^{p}\subseteq B_{1}^{q }\subseteq B_{c}^{p}$?

A: For one fixed $n$?  Yes.  All norms on $\mathbb R^n$ compatible with the usual topology are "equivalent" in this sense.  (answers both 1 and 2)
The smallest such $c$, offhand I don't know.  But would consult perhaps Hardy, Littlewood, Polya, Inequalities.
