内接于任意平面区域的胖三角形
Fat triangles inscribed in arbitrary planar domains
AI总结：
该研究证明平面任意有界开集存在内角均大于约0.3度的一致胖内接三角形，同时构造出所有内接三角形均有角小于约55度的多边形，揭示存在介于0到60度间的未知临界角Θ。
摘要：
1964年，A. Bruckner发现平面上任意有界开集都存在一个内接三角形，即该三角形包含于开集内且三个顶点都位于边界上。我们证明了这个三角形可以取到一致胖的形式，更确切地说，其所有内角都大于约0.3度，且该性质与初始开集的选取无关。我们还构造了一个多边形，其中所有内接三角形都不是太胖的，即至少有一个角小于约55度。这些结果表明存在一个严格介于0和60之间的最大值Θ（其精确值仍未知），使得所有有界开集都存在一个所有内角均大于等于Θ度的内接三角形。
英文摘要：
In 1964 A. Bruckner observed that any bounded open set in the plane has an inscribed triangle, that is a triangle contained in the open set and with the vertices lying on the boundary. We prove that this triangle can be taken uniformly fat, more precisely having all internal angles larger than $\sim 0.3$ degrees, independently of the choice of the initial open set. We also build a polygon in which all the inscribed triangles are not too-fat, meaning that at least one angle is less than $\sim 55$ degrees. These results show the existence of a maximal number $Θ$ strictly between 0 and 60, whose exact value remains unknown, for which all bounded open sets admit an inscribed triangle with all angles larger than or equal to $Θ$ degrees.
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