Limits of equi-affine equi-distant loci of planar convex domains with two non-parallel asymptotes

TL;DR

This paper introduces equi-affine invariants for convex domains using tropical structures, proves a related theorem for unbounded domains with two non-parallel asymptotes, and provides an explicit mean value formula for the unit disk.

Contribution

It develops a new method to define equi-affine invariants via tropical structures and proves a key theorem for unbounded convex domains with specific asymptotes.

Findings

Proves an analogue of the conjectured limit for unbounded domains with two non-parallel asymptotes.

Provides an explicit formula for the mean value at the center of the unit disk.

Introduces equi-affine invariants based on averaging over tropical structures.

Abstract

In this note, we introduce equi-affine invariants by averaging over the space of tropical structures of fixed covolume. Applied to the tropical distance series, this construction produces a family of equi-affine invariant functions associated with convex domains which are expected to satisfy a number of remarkable properties. We conjecture a limiting description of the associated level sets in the compact case, and we prove an analogue of this statement for unbounded domains with two non-parallel asymptotes. In addition, we give an explicit formula for the arithmetic mean value at the center of the unit disk.