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

Nikita Kalinin, Mikhail Shkolnikov

TL;DR

The paper introduces the equi-affine distance function $\mathcal{A}_{h}^{\Omega}$ by averaging the tropical distance $\mathcal{F}_{\Lambda}^{\Omega}$ over co-area-one tropical structures, yielding an equi-affine invariant tool for planar convex domains. It proves a hyperbola-limit theorem for domains with two non-parallel asymptotes, showing that level sets converge to a hyperbola branch after scaling, and it conjectures that compact domains exhibit an ellipsoidal limit in the same framework. The methodology relies on $\operatorname{SL}_2(\mathbb{R})$ symmetries and a split Cartan subgroup to reduce to the quadrant, where explicit forms like $\mathcal{A}_{h}^{\rightangle}(x,y)=c_h\sqrt{xy}$ arise, and it includes a concrete computation for the center of the unit disk. These results tie tropical averaging to classical affine geometry, hinting at connections to the Mahler conjecture and offering a pathway to higher-dimensional generalizations and links with affine curvature flows.

Abstract

In this note we discuss the novel approach to define equi-affine invariants starting from tropical geometry, where one averages over the space of tropical structures of fixed co-volume. Applied to the tropical distance series, this gives a family of equi-affine invariant functions associated with convex domains which are expected to satisfy a number of remarkable properties. The present note contains a conjecture about the limit structure of corresponding level sets in the compact case, as well as the proof of an analogue of this conjecture for unbounded domains with two non-parallel asymptotes. In addition, we carry out an explicit computation for the arithmetic mean value at center of the unit disk.

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

TL;DR

The paper introduces the equi-affine distance function by averaging the tropical distance over co-area-one tropical structures, yielding an equi-affine invariant tool for planar convex domains. It proves a hyperbola-limit theorem for domains with two non-parallel asymptotes, showing that level sets converge to a hyperbola branch after scaling, and it conjectures that compact domains exhibit an ellipsoidal limit in the same framework. The methodology relies on symmetries and a split Cartan subgroup to reduce to the quadrant, where explicit forms like arise, and it includes a concrete computation for the center of the unit disk. These results tie tropical averaging to classical affine geometry, hinting at connections to the Mahler conjecture and offering a pathway to higher-dimensional generalizations and links with affine curvature flows.

Abstract

In this note we discuss the novel approach to define equi-affine invariants starting from tropical geometry, where one averages over the space of tropical structures of fixed co-volume. Applied to the tropical distance series, this gives a family of equi-affine invariant functions associated with convex domains which are expected to satisfy a number of remarkable properties. The present note contains a conjecture about the limit structure of corresponding level sets in the compact case, as well as the proof of an analogue of this conjecture for unbounded domains with two non-parallel asymptotes. In addition, we carry out an explicit computation for the arithmetic mean value at center of the unit disk.
Paper Structure (5 sections, 4 theorems, 9 equations, 2 figures)

This paper contains 5 sections, 4 theorems, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definition, conjecture, theorem
  3. The proof
  4. Example: center of the disk
  5. Discussion

Key Result

Theorem 1

Let $\Omega$ be a convex planar domain with two non-parallel asymptotes. Denote by $H^\Omega$ a branch of a hyperbola with the same asymptotes. Then, for any $h>0$ the classes of level sets $[(\mathcal{A}_{h}^\Omega)^{-1}(t)]\in\mathcal{C}$ converge to $[H^\Omega]$ as $t\rightarrow+\infty.$

Figures (2)

  • Figure 1: An illustration for Lemma.
  • Figure 2: A plot of $\mathcal{A}^{\square}_1$ for the square $\square=[-1,1]^2\subset\mathbb{R}^2.$

Theorems & Definitions (8)

  • Definition
  • Conjecture
  • Theorem
  • Lemma
  • proof
  • Corollary
  • Corollary
  • proof : Proof of the Theorem