Stabbing Convex Bodies with Lines and Flats

$\\newcommand{\\eps}{\\varepsilon}\\newcommand{\\tldO}{\\widetilde{O}}$We study the problem of constructing weak $\\eps$-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 $[0,1]^d\\bigr.$. Specifically, a $(k,\\eps)$-net is a set of $k$-flats, such that any convex body in $[0,1]^d$ of volume larger than $\\eps$ is stabbed by one of these $k$-flats. We show that for $k \\geq 1$, one can construct $(k,\\eps)$-nets of size $O(1/\\eps^{1-k/d})$. We also prove that any such net must have size at least $\\Omega(1/\\eps^{1-k/d})$. As a concrete example, in three dimensions all $\\eps$-heavy bodies in $[0,1]^3$ can be stabbed by $\\Theta(1/\\eps^{2/3})$ lines. Note, that these bounds are sublinear in $1/\\eps$, and are thus somewhat surprising.