最优部分板覆盖
Optimal partial plank coverings
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中文总结 AI 辅助
该研究针对固定总宽度的板，解决凸体部分覆盖的最优放置问题，证明欧几里得球和平面凸体的最优部分覆盖均可由单个板实现。
中文摘要 AI 辅助
欧几里得空间中，宽度为$w$的板是位于两个间距为$w$的平行超平面之间的点集。Bang定理指出，若一组板覆盖凸体$K$，则它们的总宽度至少为$K$的宽度，即包含$K$的最薄板的宽度。我们研究该问题的定量变体：当板的总宽度固定时，如何放置板以尽可能多地覆盖$K$的体积。对于$K$为欧几里得球的核心情况，Károly Bezdek询问最优排列是否为以原点为中心的单个板，我们对此给出肯定回答。我们还证明，对于每个平面凸体，最优部分覆盖可由单个板实现。
英文摘要
A plank of width $w$ in a Euclidean space is the set of points lying between two parallel hyperplanes at distance $w$ from each other. Bang's theorem says that if a family of planks covers a convex body $K$, then their total width is at least the width of $K$, that is, the width of the thinnest plank containing $K$. We study a quantitative variant of this problem in the case where the total width of the planks is fixed. How should the planks be placed so as to cover as much of the volume of the body as possible? For the central case where $K$ is a Euclidean ball, Károly Bezdek asked whether the optimal arrangement consists of a single plank centered at the origin. We give an affirmative answer to this question. We also show that for every planar convex body an optimal partial covering is attained by a single plank.
