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

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.