There is the question,Because when we consider the gromov-hausdorff distance,we must fix the metric,so we use the natural metric induced from the embedding $S_n \to R_{n+1}$.Is it possible for us to compute the gromov-hausdorff distance $d_{G-H}(S_n,S_m)$ for two different spheres $S_n$ and $S_m$,$m\neq n$?

for example if we want to calculate $d_{G-H}(S_2,S_3)=\inf_{M,f,g}d_{M}(S_2,S_3)$,where $M$ run in all possible metric space.and $f:S_2\to M$ is all possible isometry stay the distance,not only the metric.so do $g$.

at least wen can embedding $S_2$,$S_3$ into $R^3$ in a canonical way.this will lead to a upon bound:$g_{G-H}(S_2,S_3)\leq \sqrt{3}$.and in general case we have $d_{G-H}(S_m,S_n)\leq \sqrt{max(m,n)}$.but it is difficult to get a lower bound control for me.because we need to take the inf in all possible metric space $M$.especially I conjecture $d_{G-H}(S_m,S_n)\geq \lambda_{m,n}\sqrt{max(m,n)}$,where $\liminf_{m,n\to \infty}\lambda_{m,n}>0$.

I only know the knowledge of gromov-hausdorff from Peterson's Riemann  Geometry.Unfortunately there is not enough information to compute the gromov-hausdorff distance,so this problem may be very stupid,I will be very appreciate to the pointer.