Banach-Mazur distance between [$P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$](https://arxiv.org/pdf/1707.04830.pdf) where $P_n$ is regular polygon in $n$ sides. Do closed form or approximate results exist (at least at special infinitely many $n,k$) for

1. $d(P_n,P_k)$?

2. $d(P_n^{\otimes t},P_k^{\otimes t})$ where $\otimes t$ is [tensor product](https://mathoverflow.net/questions/115677/is-the-tensor-product-of-polyhedra-a-polyhedron)?

3. $d(P_n^{\times t},P_k^{\times t})$ where $\times t$ is [cartesian product](https://en.wikipedia.org/wiki/Hanner_polytope)?

4. distances between Minkowski sums of polygons?

5. distances between Unions of polygons (with right modification of distance)?

At least do we know a situation with regular polygons where the distance becomes transcendental?