Equilateral dimension of the planar Banach--Mazur compactum

TL;DR

The paper proves that the planar Banach--Mazur compactum contains arbitrarily large finite equilateral sets by constructing convex bodies $K_a$ from a regular $4N$-gon and the unit disk, encoded by balanced symmetric binary sequences. For distinct sequences $a,b$ in a balanced family, the Banach--Mazur distance $d_{BM}(K_a,K_b)$ equals $d_N^2$ with $d_N=\frac{1}{\cos \frac{\pi}{4N}}$, yielding an equilateral set of size at least $C^N$, where $C=(\frac{2^{18}}{2^{18}-1})^{\frac{1}{20}}>1$. The construction is shown to be essentially optimal in order, matching Bronstein’s bounds for $(1+\varepsilon)$-separated sets in dimension $2$, and the proofs combine a careful operator-perturbation analysis with a probabilistic selection of balanced sequences. The work opens questions about higher-dimensional extensions and continuous-distance regimes, and suggests a potential universality-like behavior of the planar BM-compactum for finite metric configurations.

Abstract

We prove that there are arbitrarily large equilateral sets of planar and symmetric convex bodies in the Banach--Mazur distance. The order of the size of these $d$-equilateral sets asymptotically matches the bounds of the size of maximum-size $d$-separated sets (determined by Bronstein in 1978), showing that our construction is essentially optimal.

Figures (4)

• Figure 1: An example of two convex bodies $K_a$ (colored in blue) and $K_b$ (colored in red), which arise as certain combinations of the regular $12$-gon with the Euclidean unit disc. For the purpose of clarity of the drawing, an illustration is shown for small $N$ equal to $3$. In this case, the presented convex bodies are not in balance according to our definition. Nevertheless, looking at the dashed ray through $w_1$ we see that $r=d_N$ is the smallest positive dilatation factor such that $K_a \subseteq rK_b$. The same holds true for the inclusion $K_b \subseteq rK_a$, as shown by the dotted line passing trough $w_3$.
• Figure 2: In the region between $x_j$ and $x_{j+1}$ the convex body $K_a$ is circular (colored in blue), while $K_c$ is polygonal (colored in red). Since $\beta \in \left [-\frac{\pi}{4N}, \frac{\pi}{4N} \right ]$, the ray through $\widetilde{U}(w_j)$ is still in this region and the ratio between points from $\mathop{\mathrm{bd}}\nolimits K_a$ and $\mathop{\mathrm{bd}}\nolimits U(K_b)$ on this ray is equal to $d_N$.
• Figure 3: In the region between $x_i$ and $x_{i+1}$ the convex body $K_a$ is polygonal (colored in blue), while $K_c$ is circular (colored in red). Since $\beta \in \left [-\frac{\pi}{4N}, \frac{\pi}{4N} \right ]$, the ray through $\widetilde{U}^{-1}(w_i)$ is still in this region, so that $d_Nw_i=\widetilde{U}(\widetilde{U}^{-1}(d_N w_i) \in \mathop{\mathrm{bd}}\nolimits U(K_b)$ and the ratio between points from $\mathop{\mathrm{bd}}\nolimits U(K_b)$ and $K_a$ on the ray through $w_i$ is equal to $d_N$.
• Figure 4: In the region between $x_{i_2-1}$ and $x_{i_2+2}$ the convex body $K_a$ is polygonal (colored in blue), while $K_c$ is circular (colored in red). Since the operator $P$ does not change the direction of any line by more than $\frac{\pi}{2N}$, and $\beta \in \left [-\frac{\pi}{4N}, \frac{\pi}{4N} \right ]$, the ray $\ell_2$ intersecting the unit circle in $y$ is still in the region between $x_{i_2-1}$ and $x_{i_2+2}$. It follows that $z = P(\widetilde{U}(y))$ is a boundary point of $P(U(K_b))$, so that the ratio between points from $\mathop{\mathrm{bd}}\nolimits P(U(K_b))$ and $K_a$ on the ray $\ell$ is at least $d_N$.

Theorems & Definitions (9)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['mainthm']}
• Remark 5