Quadratic Crofton and sets that see themselves as little as possible

Monatshefte für Mathematik Pub Date : 2024-01-29 DOI:10.1007/s00605-023-01934-y

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Abstract

Let $\Omega \subset \mathbb {R}^2$ and let $\mathcal {L} \subset \Omega $ 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 $L \le \text{ diam }(\Omega )$ . 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 $\Omega $ 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.

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二次方克罗夫顿和尽可能少看到自己的集合

Abstract Let $\Omega \subset \mathbb {R}^2$ and let $\mathcal {L} \subset \Omega $ be a one-dimensional set with finite length $L =|\mathcal {L}|$ .我们感兴趣的是一个能量函数的最小值，这个函数测量的是一个集合在所有方向上投影到自身的大小：因此，我们要求的是集合尽可能小地看到自身（适当地解释）。该函数的最小值显然是直线的子集，但这只有在 $L \le \text{ diam }(\Omega )$ 时才有可能。这个问题有一个等价的表述：随机直线与 $\mathcal {L}$的预期交点数只取决于 $\mathcal {L}$的长度（克罗夫顿公式）。我们感兴趣的是$\mathcal {L}$集，它能使预期交点数的方差最小化。我们解决了凸（\ω \）和略小于所有 L 值一半的问题：在那里，最小化集合是边界副本和线段的结合。

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Monatshefte für Mathematik

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