Given two centered planar n-point sets represented by complex numbers x_1,…,x_n,y_1,…,y_n∈C with ∑_i=1^n x_i=∑_i=1^n y_i=0, define the permutation polygon P(X,Y)=operatornameconvz_sigma:sigma∈ S_n, where z_sigma=∑_i=1^n overlinex_iy_sigma(i). Determine the maximum possible number of vertices of P(X,Y) as a function of n. The claimed sharp answer is n(n-1) for every n≥ 2: every such polygon has at most n(n-1) vertices, and equality is attainable.

Rote’s rotation-assignment open problem

Given two centered planar $n$ -point sets represented by complex numbers $x_1,\ldots,x_n,y_1,\ldots,y_n\in\mathbb{C}$ with $\sum_{i=1}^n x_i=\sum_{i=1}^n y_i=0$ , define the permutation polygon $P(X,Y)=\operatorname{conv}\left\{z_\sigma:\sigma\in S_n\right\}$ , where $z_\sigma=\sum_{i=1}^n\overline{x_i}y_{\sigma(i)}$ . Determine the maximum possible number of vertices of $P(X,Y)$ as a function of $n$ . The claimed sharp answer is $n(n-1)$ for every $n\ge 2$ : every such polygon has at most $n(n-1)$ vertices, and equality is attainable.

References

Primary source

arXiv

Additional references

Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry — arXiv — Subhransu S. Bhattacharjee, Dylan Campbell, Rahul Shome

combinatorial-optimization computational-geometry

Posted 1 hour ago · edited 1 hour ago

Progress summary

Refreshed 1 hour ago

Claimed solved

A new paper claims to solve Rote’s problem by giving an exact bound and a global reconstruction method, but the claim has not been independently checked.

Rote’s problem is a specialized combinatorial-geometry question about determining the maximum number of relevant polygon vertices and replacing local optimization with a global assignment procedure.

October 2026 global-solution claim

Bhattacharjee, Campbell, and Shome’s paper Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry claims the exact maximum and an assignment-based reconstruction algorithm, which would resolve the problem. The proof and implementation have not been independently assessed.

Current status (as of October 2026): A complete solution is claimed in a recent paper, but the result remains unverified; absent confirmation, the problem should be treated as open.

Sources

No solutions have been posted yet.