For fixed dimension $d$ and large $R$ consider all non-zero integer vectors in the ball $B(0,R)\subset \mathbb{R} ^d$ of radius $R$ centered at the origin. The number of such vectors grows as $c_d\cdot R^d$, as well as the number of distinct 1-dimensional linear spaces they generate (with a different constant, of course, both constant are well-known).
Is it correct that they generate (at least) $c(k, d)\cdot R^{kd}$ distinct $k$-dimensional planes for all $k=1,2,\ldots,d-1$ with some positive constants $c(k, d)$?
A connected question: is it correct that the hyperplanes orthogonal to these vectors partition the space onto at least $C(d)\cdot R^{d(d-1)}$ open regions?
If yes (what I expect to be the case and possibly even have a proof, but quite tedious and with unpleasant details), are the sharp constants computed and what are the simplest proofs?
