Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ 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?

3. $d(P_n^{\times t},P_k^{\times t})$ where $\times t$ is cartesian product?

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?