Is it possible to place a sphere $S^m$ and another sphere $S^n$ in Euclidean $k$-dimensional space $R^k$ in such a way that the distance from any point of the first sphere to any point of the second sphere be constant?
For the case $m=n=0$ it is obviously possible for $k=2$: the solution is to place the two points of the second $0$-sphere as the two vertices of a rhombus with the other two vertices being the two points of the first $0$-sphere. Here we observe that $k=m+n+2$. Notice that the centres of two "spheres" coincide and the diameters are orthogonal.
For the case $m=1,n=0$, again, the solution is obvious and requires $k=3$: you just have to make the two centres coincide and make the diameter of the $0$-sphere to be orthogonal to all diameters of the $1$-sphere. Again, $k=m+n+2$, which suggests a conjecture that it holds for all $m,n$.
But how to solve this for arbitrary $m$ and $n$?