SIGMA 22 (2026), 071, 10 pages      arXiv:2512.21154      https://doi.org/10.3842/SIGMA.2026.071

Limits of Equi-Affine Equi-Distant Loci of Planar Convex Domains with Two Non-Parallel Asymptotes

Nikita Kalinin ^ab and Mikhail Shkolnikov ^c
a) Guangdong Technion Israel Institute of Technology (GTIIT), 241 Daxue Road, Shantou, Guangdong Province 515603, P.R. China
^b) Technion-Israel Institute of Technology, Haifa, 32000, Haifa district, Israel
^c) Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences, Akad. G. Bonchev St, Bl. 8, 1113 Sofia, Bulgaria

Received December 25, 2025, in final form July 29, 2026; Published online August 08, 2026

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, showing the universality of the affine curvature of the resulting limiting hyperbola. In addition, we give an explicit formula for the arithmetic mean value at the center of the unit disk.

Key words: equi-affine geometry; tropical distance; convex domains; unimodular lattices.

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