On the plane, one can give a simple condition for the intersection of two circles $\mathcal{C}_1(O_1,r_1),\mathcal{C}_2(O_2,r_2)$ : they intersect iff $|r_1-r_2|\le|O_1O_2|\le r_1+r_2$.
Likewise, can we give such a (geometrical) definition for the intersection of two circles on $\Bbb{R}^n,n\ge2$ ?
More generally, if we call $A_p$ the equivalent of a circle on $R^p$ (a sphere on $R^3$ and so on), on what geometrical condition do two such shapes intersect on $R^n$ ?
Could we even generalize to the intersection of $A_p,A_q$ on $R^n$ ?
For two spheres of maximal dimensions the condition is the very same as for circles in the plane (actually, the case $n=1$ is a weird exception to this rule).
For sub-dimensional spheres, note that those "live" in hyperplanes accordingly. If the intersection of these hyperplanes is at least two-dimensional and the radius bound holds, then we again have an intersection. If the intersection of the hyperplanes is one-dimensional, the spheres may at most share a single point (which requires $|O_1O_2|=|r_1-r_2|$ or $|O_1O_2|=r_1+r_2$); note however that the spheres may be linked without intersection. If the intersection of hyperplanes is a single point, this may be a point of intersection of the spheres (if it actually belongs to both of course). Finally, if the hyperplanes are disjoint then so are the spheres.