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arXiv · 2512.21154

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

Nikita Kalinin·Mikhail Shkolnikov
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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.

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Nikita Kalinin, Mikhail Shkolnikov. 2025-12-24. Limits of Equi-Affine Equi-Distant Loci of Planar Convex Domains with Two Non-Parallel Asymptotes. https://doi.org/10.3842/sigma.2026.071

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