Finding balls that contain a point Suppose you are given a collection of $n$ balls in $\mathbb{R}^d$, and you want to preprocess them in such a way that you can later query them to find all spheres which contain any test point.

What efficient solutions are known for this problem?

The trivial answer is to just do a linear scan over all the spheres, which takes no preprocessing time but requires an overhead of $O(n)$ per query.

In 1D you can solve the problem using an interval tree, giving $O(\log(n) + k)$ query time with $O(n \log(n))$ preprocessing and $O(n)$ space, but it isn't clear to me that this generalizes.

I suspect there should be some sort of reduction here, since I can't seem to find any discussion of this problem in the literature which leads to me to think that it is trivial (though I am probably searching for the wrong terms). 

There is a reduction to metric range searching in hyperbolic geometry with the signature $(+++ ... -)$, but I don't know how to do hyperbolic range searching efficiently (so it isn't much help).

If this question turns out to be too easy, you can also consider the situation where the set of balls is dynamic.
Note that point location in an arrangement of balls (of different radii) is equivalent to point location in an arrangement of hyperplanes one dimension higher by using the usual paraboloid lifting map $(x_1, \ldots, x_d) \mapsto (x_1, \ldots, x_d, \sum_i x_i^2)$. If you call this lifting map $\Phi$, then it's easy to see that $p \in B$ if and only if $\Phi(p)$ is above $\Phi(B)$. 

Now you have the problem of point location in an arrangement of hyperplanes (note that once you have such a structure it's usually easy to annotate each cell with the list of hyperplanes it is above). That problem can be solved by a variety of methods: for example there's an algorithm by Meiser that runs in time $O(d^5 \log n)$ but requires $n^d$ space. It's not clear from the question which is more important: space or time...