论文标题
二次crofton，并将自己尽可能少
Quadratic Crofton and sets that see themselves as little as possible
论文作者
论文摘要
令$ω\ subset \ mathbb {r}^2 $，让$ \ mathcal {l} \ subsetω$是有限长度$ l = | \ m atarcal {l} | $的一维集合。我们对一种能量功能的最小化感兴趣，该功能可以衡量在各个方向上投射到自身的集合的大小：因此，我们正在要求将自己视为尽可能少的集合（适当解释）。功能的显而易见的最小化是直线的子集，但这仅适用于$ l \ leq \ mbox {diam}（ω）$。该问题具有等效的公式：随机线和$ \ Mathcal {l} $之间的相交数量的预期数仅取决于$ \ Mathcal {l} $（Crofton的公式）的长度。我们对集合$ \ MATHCAL {l} $感兴趣，该$ \ Mathcal {l} $最大程度地减少了预期的交叉点的差异。我们解决了凸面$ω$的问题，略低于$ l $的所有值的一半：在那里，最小化的集合是边界和线段的副本结合。
Let $Ω\subset \mathbb{R}^2$ and let $\mathcal{L} \subset Ω$ be a one-dimensional set with finite length $L =|\mathcal{L}|$. We are interested in minimizers of an energy functional that measures the size of a set projected onto itself in all directions: we are thus asking for sets that see themselves as little as possible (suitably interpreted). Obvious minimizers of the functional are subsets of a straight line but this is only possible for $L \leq \mbox{diam}(Ω)$. The problem has an equivalent formulation: the expected number of intersections between a random line and $\mathcal{L}$ depends only on the length of $\mathcal{L}$ (Crofton's formula). We are interested in sets $\mathcal{L}$ that minimize the variance of the expected number of intersections. We solve the problem for convex $Ω$ and slightly less than half of all values of $L$: there, a minimizing set is the union of copies of the boundary and a line segment.
