论文标题
凸内部旋转Kakeya套件
Rotation inside convex Kakeya sets
论文作者
论文摘要
令$ k $为$ \ mathbb {r}^d $中的凸形主体（紧凑型凸组），其中包含每个可能的方向的另一个正文$ s $的副本。是否总是可以将$ s $的任何一个副本不断地移动到另一个$ k $的内部？作为一个更强大的问题，是否总是有可能在每个方向上连续选择一个$ s $的副本？这些问题是克罗夫特问的。 我们表明，在两个维度上，更强大的问题总是有一个肯定的答案。我们还表明，在三个维度上，答案是负面的，即使对于$ s $是线段的情况下，在任何维度上，第一个问题在$ s $是线条段时都会有正面答案。而且我们证明，令人惊讶的是，第一个问题的答案是一般$ S $的维度四。
Let $K$ be a convex body (a compact convex set) in $\mathbb{R}^d$, that contains a copy of another body $S$ in every possible orientation. Is it always possible to continuously move any one copy of $S$ into another, inside $K$? As a stronger question, is it always possible to continuously select, for each orientation, one copy of $S$ in that orientation? These questions were asked by Croft. We show that, in two dimensions, the stronger question always has an affirmative answer. We also show that in three dimensions the answer is negative, even for the case when $S$ is a line segment -- but that in any dimension the first question has a positive answer when $S$ is a line segment. And we prove that, surprisingly, the answer to the first question is negative in dimension four for general $S$.
