热带测度、各向异性等周不等式与蜂窝
Tropical measures, anisotropic isoperimetric inequality and honeycomb
AI总结：
本文引入了基于热带度量的热带球面测度，并证明了二维中的精确热带等周不等式，推导出热带蜂窝定理，同时探讨了热带Minkowski内容和相关几何结构。
摘要：
我们介绍了基于热带度量的热带球面测度，其是球面Hausdorff测度的类比。该测度具有平移不变性，但不同于勒贝格测度，它不具有旋转或反射不变性。它在n维（但不包括k维，k < n）可测子集上与勒贝格测度一致，并在可测曲线上恢复热带长度。在二维情况下，我们证明了一个精确的热带等周不等式，等号成立仅当为热带盘时，并推导出热带蜂窝定理。我们还引入了热带的Minkowski内容类比，并显示热带球是相应的Wulff形状。这导致了热带等周问题的各向异性类型，并因此在R^n中推导出热带蜂窝定理。最后，我们描述了热带对偶范数和对偶球，比较了热带球面和Minkowski表面测度，并证明它们在平面和R^n中具有平行于热带球或其对偶的面的多面体上一致。
英文摘要：
We introduce a tropical spherical measure on $\mathbb{R}^n$ that is based on the tropical metric and is an analogue of spherical Hausdorff measure. This measure is translation invariant but, unlike Lebesgue measure, is not invariant under rotations or reflections. It agrees with Lebesgue measure on $n$-dimensional (but not on $k$-dimensional, $k<n$) measurable subsets of $\mathbb{R}^n$, and on rectifiable curves it recovers tropical length. In dimension $2$ we prove a sharp tropical isoperimetric inequality, with equality precisely for tropical disks, and deduce a tropical honeycomb theorem. We also introduce a tropical analogue of Minkowski content and show that the tropical ball is the associated Wulff shape. This yields an anisotropic type of the tropical isoperimetric problem and consequently a tropical honeycomb theorem in $\mathbb{R}^n$. Finally, we describe the tropical dual norm and dual ball, compare the tropical spherical and Minkowski surface measures, and prove that they agree in the plane and on polytopes in $\mathbb{R}^n$ whose facets are parallel to facets of the tropical ball or its dual.
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