Does Banach-Mazur distance between regular polygons admit any structure that lends to approximation or exact results in particular situations?

Question

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?

Answer 1

This is really more of a comment than an answer because it only answers your last question, but the argument is a little too long.

The answer to your last question is that the Banach-Mazur distance between regular polygons can never be transcendental. This comes from the fact that the Banach-Mazur distance between regular polygons is definable in the theory of real closed fields, $\mathrm{RCF}$, as well as the fact that $\mathrm{RCF}$ is O-minimal. Another consequence is that since $\mathrm{RCF}$ is decidable there is a uniform algorithm for computing the Banach-Mazur distances between regular polygons, although the naive algorithm is extremely slow.

For a precise statement, fix a subfield $K\subseteq \mathbb{R}$ and assume that $K$ is real closed, i.e. closed under square roots of positive numbers and containing all roots of odd ordered polynomials. What we have is the following (using the notation of the paper you linked):

Proposition: If $\mathscr{A}$ and $\mathscr{B}$ are convex bodies in $\mathbb{R}^n$ defined by finite families of rational inequalities with coefficients in $K$, then $d(\mathscr{A},\mathscr{B})\in K$.

Proof: An affine transformation (and in particular a homothety) is specified by a finite list of real numbers and the action of it on an element of $\mathbb{R}^n$ is defined by some polynomial equations (in particular linear), so the following is a formula in $\mathrm{RCF}$,

$$\varphi(\lambda,\overline{a},\overline{b})\equiv\exists T \exists h_\lambda (T(\mathscr{A})\subset \mathscr{B} \subset h_\lambda(T(\mathscr{A})),$$

where $\overline{a}$ and $\overline{b}$ are lists of the parameters needed to define $\mathscr{A}$ and $\mathscr{B}$, $\exists T$ means 'exists an affine transformation $T$', and $\exists h_\lambda$ means 'exists a homothety with ratio $\lambda$'. Note that when you interpret $\varphi$ in $K$, what these quantifiers really mean is 'exists an affine transformation with coefficients in $K$'.

Now I claim that for any real closed field $K\subseteq \mathbb{R}$ (although actually this would work for any subfield of $\mathbb{R}$), we have that $d(\mathscr{A},\mathscr{B})=\inf\{\lambda:K\models\varphi(\lambda,\overline{a},\overline{b})\}$. This is because any given affine transformation/homothety can be approximated arbitrarily well by one with coefficients in $K$ (specifically this approximation is uniform on bounded sets, which is all we need because $\mathscr{A}$ and $\mathscr{B}$ are compact).

Now, by O-minimality, every definable subset of $K$ is a finite union of intervals (possibly infinite or zero length), so in particular every definable set in $K$ that has an infimum has an infimum in $K$, so we have that $d(\mathscr{A},\mathscr{A})$ is in $K$, as required. $\square$

In particular, every regular polygon is definable using linear inequalities with algebraic coefficients, so the BM distances between regular polygons must be algebraic.

I think with a little work you can show that all the other polytopes featured in your question are definable in $\mathrm{RCF}$, which would imply that all those BM distances are algebraic as well.