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$\newcommand{\eps}{\varepsilon}\newcommand{\tldO}{\widetilde{O}}$Consider theproblem of constructing weak $\eps$-nets where the stabbing elements are linesor $k$-flats instead of points. We study this problem in the simplest settingwhere it is still interesting -- namely, the uniform measure of volume over thehypercube $[0,1]^d\bigr.$. 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 canconstruct $(k,\eps)$-nets of size $O(1/\eps^{1-k/d})$. We also prove that anysuch net must have size at least $\Omega(1/\eps^{1-k/d})$. As a concreteexample, in three dimensions all $\eps$-heavy bodies in $[0,1]^3$ can bestabbed by $\Theta(1/\eps^{2/3})$ lines. Note, that these bounds are\emph{sublinear} in $1/\eps$, and are thus somewhat surprising. The newconstruction also works for points providing a weak $\eps$-net of size$O(\tfrac{1}{\eps}\log^{d-1} \tfrac{1}{\eps} )$.

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