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 $\lambda\geq 1$, the critical covering area $A^*(\lambda)$ is the minimum value for which any set of disks with total area at least $A^*(\lambda)$ can cover a rectangle of dimensions $\lambda\times 1$. We show that there is a threshold value $\lambda_2 = \sqrt{\sqrt{7}/2 - 1/4} \approx 1.035797\ldots$, such that for $\lambda<\lambda_2$ the critical covering area $A^*(\lambda)$ is $A^*(\lambda)=3\pi\left(\frac{\lambda^2}{16} +\frac{5}{32} + \frac{9}{256\lambda^2}\right)$, and for $\lambda\geq \lambda_2$, the critical area is $A^*(\lambda)=\pi(\lambda^2+2)/4$; these values are tight. For the special case $\lambda=1$, i.e., for covering a unit square, the critical covering area is $\frac{195\pi}{256}\approx 2.39301\ldots$. The proof uses a careful combination of manual and automatic analysis, demonstrating the power of the employed interval arithmetic technique.

中文翻译：

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最坏情况下用磁盘对矩形进行最优覆盖

我们通过给出矩形的最坏情况最优盘覆盖的完整表征，为几何优化的基本问题提供了解决方案：对于任何 $\lambda\geq 1$，临界覆盖区域 $A^*(\lambda)$ 是总面积至少为 $A^*(\lambda)$ 的任何一组磁盘可以覆盖尺寸为 $\lambda\times 1$ 的矩形的最小值。我们证明存在一个阈值 $\lambda_2 = \sqrt{\sqrt{7}/2 - 1/4} \approx 1.035797\ldots$，使得对于 $\lambda<\lambda_2$ 的临界覆盖区域 $A ^*(\lambda)$ 是 $A^*(\lambda)=3\pi\left(\frac{\lambda^2}{16} +\frac{5}{32} + \frac{9}{ 256\lambda^2}\right)$，对于$\lambda\geq\lambda_2$，临界区为$A^*(\lambda)=\pi(\lambda^2+2)/4$；这些值是严格的。对于特殊情况 $\lambda=1$，即覆盖一个单位正方形，关键覆盖区域是 $\frac{195\pi}{256}\approx 2.39301\ldots$。证明使用了手动和自动分析的仔细组合，展示了所采用的区间算术技术的威力。