There are several related and very interesting problems and theorems:  

* [Schinzel's theorem](http://mathworld.wolfram.com/SchinzelsTheorem.html) - solves the problem in $\mathbb{R}^2$ using so-called [Schinzel circle](http://mathworld.wolfram.com/SchinzelCircle.html). It seems intuitively clear that it generalizes to higher dimensions by slightly adjusting radius of a hypersphere so that it contains exactly the same lattice points as its section in lower dimension, but of course, a rigorous proof would be of interest (and, probably, already published somewhere)
* [Kulikowski's theorem](http://mathworld.wolfram.com/KulikowskisTheorem.html) - gives explicit construction in $\mathbb{R}^3$

And similar problems related to interior points:

* [Steinhaus' theorem](http://mathworld.wolfram.com/CircleLatticePoints.html)
* [Browkin's theorem](http://mathworld.wolfram.com/BrowkinsTheorem.html)