{"title":"Quadratic Crofton and sets that see themselves as little as possible","authors":"","doi":"10.1007/s00605-023-01934-y","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Let <span> <span>\\(\\Omega \\subset \\mathbb {R}^2\\)</span> </span> and let <span> <span>\\(\\mathcal {L} \\subset \\Omega \\)</span> </span> be a one-dimensional set with finite length <span> <span>\\(L =|\\mathcal {L}|\\)</span> </span>. 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 <span> <span>\\(L \\le \\text{ diam }(\\Omega )\\)</span> </span>. The problem has an equivalent formulation: the expected number of intersections between a random line and <span> <span>\\(\\mathcal {L}\\)</span> </span> depends only on the length of <span> <span>\\(\\mathcal {L}\\)</span> </span> (Crofton’s formula). We are interested in sets <span> <span>\\(\\mathcal {L}\\)</span> </span> that minimize the variance of the expected number of intersections. We solve the problem for convex <span> <span>\\(\\Omega \\)</span> </span> and slightly less than half of all values of <em>L</em>: there, a minimizing set is the union of copies of the boundary and a line segment.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-023-01934-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.
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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