It can be zero.

Take the standard metric on $\mathbb R^3$ and the one given in [this example][1] by S. Ivanov.
One can make it bit simpler in your case, but I am lazy to write it.
A similar example can be constructed in $\mathbb R^2$.

  [1]: http://mathoverflow.net/questions/125283/on-lipschitz-embeddability-of-certain-compact-metric-spaces-into-mathbbrn/125295#125295