Stabbing Convex Bodies with Lines and Flats
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We study the problem of constructing weak -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,)-net is a set of k-flats, such that any convex body in [0,1]^d of volume larger than is stabbed by one of these k-flats. We show that for k ≥ 1, one can construct (k,)-nets of size O(1/^1-k/d). We also prove that any such net must have size at least Ω(1/^1-k/d). As a concrete example, in three dimensions all -heavy bodies in [0,1]^3 can be stabbed by Θ(1/^2/3) lines. Note, that these bounds are sublinear in 1/, and are thus somewhat surprising.
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