凸Kakeya问题的推广
A Generalization of the Convex Kakeya Problem
摘要：
给定平面上的一组线段（不一定是有限的），包含每个输入线段平移的面积最小的凸区域是什么？这个问题可以被看作是一个推广的Kakeya的问题，找到一个凸区域的最小面积，使针可以旋转通过360度在这个区域内。我们表明，总是有一个最佳的区域，是一个三角形，我们给出了一个最佳的Θ（n log n）-时间算法来计算这样的三角形为一个给定的n段。我们还表明，如果目标是最小化的区域，而不是其面积的周长，然后将其中点在原点的段，并采取其凸包的结果在一个最佳的解决方案。最后，我们证明了对任意紧凸图G，G的最小包围圆盘是包含G的每个旋转副本的平移的最小周长区域。
英文摘要：
Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal Θ(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
