# Worst-Case Optimal Covering of Rectangles by Disks

We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worst-case optimal disk coverings of rectangles: For any λ≥ 1, the critical covering area A^*(λ) is the minimum value for which any set of disks with total area at least A^*(λ) can cover a rectangle of dimensions λ× 1. We show that there is a threshold value λ_2 = √(√(7)/2 - 1/4)≈ 1.035797…, such that for λ<λ_2 the critical covering area A^*(λ) is A^*(λ)=3π(λ^2/16 +5/32 + 9/256λ^2), and for λ≥λ_2, the critical area is A^*(λ)=π(λ^2+2)/4; these values are tight. For the special case λ=1, i.e., for covering a unit square, the critical covering area is 195π/256≈ 2.39301…. The proof uses a careful combination of manual and automatic analysis, demonstrating the power of the employed interval arithmetic technique.

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