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

Consider the problem of constructing weak e-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, e)-net is a set of k-flats, such that any convex body in [0, 1](d) of volume larger than e is stabbed by one of these k-flats. We show that for k = 1, one can construct (k, e)-nets of size O(1/e(1-k/d)). We also prove that any such net must have size at least O (1/e(1-k/d)). As a concrete example, in three dimensions all e-heavy bodies in [0, 1]3 can be stabbed by T(1/e(2/3)) lines. Note that these bounds are sublinear in 1/e, and are thus somewhat surprising. The new construction also works for points providing a weak e-net of size O((1/e)log(d-1)(1/e)).