A Note on Stabbing Convex Bodies with Points, Lines, and Flats

Consider the problem of constructing weak ε-nets where the stabbing elements are lines or k-flats instead of points. We study this problem in the simplest setting where it is still interesting—namely, the uniform measure of volume over the hypercube [0 , 1] ^d. Specifically, a (k, ε) -net is a set of k-flats, such that any convex body in [0 , 1] ^d of volume larger than ε is stabbed by one of these k-flats. We show that for k≥ 1 , one can construct (k, ε) -nets of size O(1 / ε^1-k/d). We also prove that any such net must have size at least Ω (1 / ε^1-k/d). As a concrete example, in three dimensions all ε-heavy bodies in [0 , 1] ³ can be stabbed by Θ (1 / ε^{2 / 3}) lines. Note that these bounds are sublinear in 1 / ε, and are thus somewhat surprising. The new construction also works for points providing a weak ε-net of size O((1 / ε) log ^d-1(1 / ε)).