Hamming cube as an L1 and L2 metric
= Hamming cube as an L1 and L2 metric
{c}
{title2=$H_n=\{0,1\}^n$}
The Hamming cube has $d_H(x,y)=\sum_j|x_j-y_j|$. Its coordinate map into $\ell_1^n$ is isometric, while the same map into $\ell_2^n$ has distance $\sqrt{d_H(x,y)}$. Thus the family of all Hamming cubes uniformly coarsely embeds into both $L^1$ and $L^2$.