Covering and packing with homothets of limited capacity
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This work revolves around the two following questions: Given a convex body C⊂ℝ^d, a positive integer k and a finite set S⊂ℝ^d (or a finite Borel measure μ on ℝ^d), how many homothets of C are required to cover S if no homothet is allowed to cover more than k points of S (or have measure larger than k)? How many homothets of C can be packed if each of them must cover at least k points of S (or have measure at least k)? We prove that, so long as S is not too degenerate, the answer to both questions is Θ_d(|S|/k), where the hidden constant is independent of d. This is optimal up to a multiplicative constant. Analogous results hold in the case of measures. Then we introduce a generalization of the standard covering and packing densities of a convex body C to Borel measure spaces in ℝ^d and, using the aforementioned bounds, we show that they are bounded from above and below, respectively, by functions of d. As an intermediate result, we give a simple proof the existence of weak ϵ-nets of size O(1/ϵ) for the range space induced by all homothets of C. Following some recent work in discrete geometry, we investigate the case d=k=2 in greater detail. We also provide polynomial time algorithms for constructing a packing/covering exhibiting the Θ_d(|S|/k) bound mentioned above in the case that C is an Euclidean ball. Finally, it is shown that if C is a square then it is NP-hard to decide whether S can be covered using |S|/4 squares containing 4 points each.
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