Checking for intersection of $d$-dimensional spheres
This is an instance of the smallest enclosing ball problem.

An equivalent statement of your problem is: Given points $x_1,\dots,x_n$, we want to determine whether there exists a point $p$ such that every point $x_i$ is at distance at most $R$ from $p$, i.e., we want to find a sphere of radius $R$ that encloses all of the points $x_1,\dots,x_n$.

So, find the smallest enclosing ball that contains all of $x_1,\dots,x_n$; then check whether the radius of that ball is at most $R$ or not.  The smallest enclosing ball problem is well-studied and you can find references and algorithms in the Wikipedia page I linked to.