Isometric embedding of an $n$-point equilateral space

Question

I'm stumped on these questions, and would appreciate a solution:

I need to find an isometric embedding of the n-point equilateral space in $l_{p}$.

And if $n=2^{d}$, an isometric embedding of the n-point equilateral space in $l^d_{\infty}$.

Answer 1

Take $n$ vectors from the standard basis of $l_p$ to find an $n$-point equilateral space in $l_p$.

How many corners does the unit cube in $l_{\infty}^d$ have?