In $d$ dimensional Euclidean space, the VC-dimension of both the set of balls and the set of half-spaces is $d+1$. It follows that the VC-dimension of balls intersected with half-spaces is $O(d \log d)$. Can we get the better result that the VC dimension of balls intersected with half-spaces is $O(d)$?
My reason for wondering is that I have seen in the appendix of several learning theory papers the claim that the VC-dimension of half-balls is at most $2d + 2$ (without proof). It seems that half-balls are not so much less powerful than arbitrary intersections of balls and half-spaces, so maybe a similar result holds here.
Thanks!