Coordinates on convex domains

Definitions and mathematical statement

Let $\Omega$ be the source's compact convex domain, let $F_\Omega$ be its assigned function, and let $C_\Omega$ be the corner locus. For a piecewise-affine function, a corner locus is the set where it is not locally affine, with vertices supplied by the associated polyhedral structure. The source-specific function and that structure are not defined in this extraction. The stated coordinate data are \[ c(\Omega)=\left(F_\Omega(v)\right)_{v\in V(C_\Omega)}. \] No finiteness, labeling, or fixed combinatorial type of this vertex set is assumed.

The source says these are complete coordinates and asks for an explicit relation between \[ c(\Omega)\quad\text{and}\quad c(\Omega^*). \] Equivalently, in a specified coordinate/moduli framework, describe the induced duality operation $\mathcal D$ with \[ c(\Omega^*)=\mathcal D(c(\Omega)). \] An abstract definition by reconstructing the domain and then dualizing would not by itself provide the explicit coordinate relation being sought. No coordinatewise pairing of vertices is presumed; duality may change the corner-locus combinatorics.

The dual-domain convention is missing. For example, if the intended planar convention is polar duality, one needs a chosen origin with $0\in\operatorname{int}\Omega$ and defines \[ \Omega^*=\{y\in{\mathbb{R}}^2:\langle x,y\rangle\le1\text{ for all }x\in\Omega\}. \] This requires the origin/normalization and is different from other projective or Legendre-dual conventions. The meaning of complete coordinates, including translation or other equivalences and vertex labeling, must be compatible with whichever duality is intended. Neither that polar convention nor a formula for $F_\Omega$ is silently substituted for the absent source definitions.

The supplied research summary reports later tropical-caustic moduli progress toward the coordinate programme but says the duality relation remains open and not fully formalized. No explicit transformation or complete follow-up citation is supplied.

Short English statement

Express the dual convex domain's corner-locus vertex values in terms of the original domain's complete coordinate data. The function, indexing/normalization, and duality convention must first be fixed.

Sources

• Local UnsolvedMath registry, permanent ID $9000001$, exact source code catalogue entry, "Coordinates on convex domains." Proposers Nikita Kalinin and Mikhail Shkolnikov.
• Kalinin and Shkolnikov, The Number pi and a Summation by SL(2,Z) (2018), section 4, Questions direction 1; https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/17-75/index.html. HTML-text extraction; recorded access 2026-07-29.
• Supplied PARTIAL-PROGRESS summary mentions a follow-up tropical-caustic moduli paper without a title, publication details, or a coordinate-duality theorem.