Limits of equi-affine equi-distant loci of planar convex domains with two non-parallel asymptotes
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| Format: | Preprint |
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2025
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| _version_ | 1866911651074146304 |
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| author | Kalinin, Nikita Shkolnikov, Mikhail |
| author_facet | Kalinin, Nikita Shkolnikov, Mikhail |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_21154 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limits of equi-affine equi-distant loci of planar convex domains with two non-parallel asymptotes Kalinin, Nikita Shkolnikov, Mikhail Mathematical Physics Algebraic Geometry Differential Geometry Number Theory 52A38, 14T05 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. |
| title | Limits of equi-affine equi-distant loci of planar convex domains with two non-parallel asymptotes |
| topic | Mathematical Physics Algebraic Geometry Differential Geometry Number Theory 52A38, 14T05 |
| url | https://arxiv.org/abs/2512.21154 |