Tropical measures, anisotropic isoperimetric inequality and honeycomb

Abstract

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.