For a given planar convex body $K$, a bisection of $K$ is a decomposition of $K$ into two closed sets $A, B$ so that $A \cap B$ is an injective continuous curve connecting exactly two boundary points of $K$. Consider a bisection of $K$ minimizing, over all bisections, the maximum diameter (resp., maximum width) of the sets in the decomposition.

In this note, we study some properties of these minimizing bisections and prove inequalities extending the classical isodiametric and isominwidth inequalities. Furthermore, we address the corresponding reverse optimization problems and establish inequalities similar to the reverse isodiametric and reverse isominwidth inequalities.

中文翻译：

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关于平面二等分的等径不等式和等宽不等式

对于给定的平面凸体 $K$，$K$ 的二等分是将 $K$ 分解为两个封闭集 $A,B$，因此 $A\cap B$ 是一条正好连接两个边界点的内射连续曲线$K$。考虑$K$ 的二等分，在所有二等分中最小化分解中集合的最大直径（相应地，最大宽度）。

在本笔记中，我们研究了这些最小化二等分的一些性质，并证明了扩展经典等径和等宽不等式的不等式。此外，我们解决了相应的反向优化问题，并建立了类似于反向等径不等式和反向等宽不等式的不等式。