发表机构
University of Szeged（塞格德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个更强的多边形弦猜想，蕴含Kuperberg四边形猜想，给出五边形情形的简短证明，并证明经典上界√2对弦平行四边形仍成立，最后讨论达到最优常数3/√5的困难。
AI 中文摘要
Kuperberg在1983年猜想：平面中的每个凸体$K$都包含在一个面积至多为$\frac{3}{\sqrt5}\\,|K|$的四边形中，极值情形为仿射正则五边形。基于数值实验，我们提出了一个更强的、纯多边形性质的猜想：每个凸多边形$P$都包含在一个面积至多为$\frac{3}{\sqrt5}\\,|P|$的四边形中，且该四边形的边平行于$P$的边或对角线。这个“弦猜想”蕴含了Kuperberg所猜想的不等式。对于五边形，这是Hong、Ismailescu、Kwak和Park的一个定理，我们通过面积恒等式给出了一个简短证明。我们还证明了经典上界$\sqrt2$对于边平行于弦的四边形仍然成立；证明方法与Ismailescu对该上界的证明类似，但出发点是一个最大面积的内接四边形，而非最小面积的外接四边形。最后，我们讨论了达到常数$\frac{3}{\sqrt5}$的困难。
英文摘要
Kuperberg conjectured in 1983 that every convex body $K$ in the plane is contained in a quadrilateral of area at most $\frac{3}{\sqrt5}\,|K|$, the extremal bodies being the affine-regular pentagons. On the basis of numerical experiments we propose a stronger conjecture of a purely polygonal nature: every convex polygon $P$ is contained in a quadrilateral of area at most $\frac{3}{\sqrt5}\,|P|$ whose sides are parallel to sides or diagonals of $P$. This \emph{chord conjecture} implies the inequality conjectured by Kuperberg. For pentagons it is a theorem of Hong, Ismailescu, Kwak and Park, of which we give a short proof by area identities. We also show that the classical bound $\sqrt2$ remains valid for quadrilaterals with sides parallel to chords; the proof is similar to Ismailescu's proof of this bound, but it starts from an inscribed quadrilateral of maximal area instead of a circumscribed quadrilateral of minimal area. We close with a discussion of the difficulties in reaching the constant $\frac{3}{\sqrt5}$.
