[Paper Review] On the Banach-Mazur Distance between the Cube and the Crosspolytope

Fei Xue|arXiv (Cornell University)|May 3, 2017

Mathematical Approximation and Integration5 references3 citations

TL;DR

This paper establishes improved explicit bounds for the Banach-Mazur distance between the $n$-dimensional cube $C_n$ and the crosspolytope $C_n^\star$, proving a lower bound of $\alpha\sqrt{n}$ with $\alpha > 1/1.71453$ and an upper bound of $(\sqrt{2}+1)\sqrt{n}$. It combines analytical estimates with computer-assisted optimization to refine prior order-$\sqrt{n}$ bounds, and conjectures tighter bounds based on numerical results in low dimensions.

ABSTRACT

In this note we study the Banach-Mazur distance between the $n$-dimensional cube and the crosspolytope. Previous work shows that the distance has order $\sqrt{n}$, and here we will prove some explicit bounds improving on former results. Even in dimension 3 the exact distance is not known, and based on computational results it is conjectured to be $\frac{9}{5}$. Here we will also present computerbased potential optimal results in dimension $4$ to $8$.

Motivation & Objective

• To improve the explicit upper and lower bounds for the Banach-Mazur distance between the $n$-dimensional cube $C_n$ and the crosspolytope $C_n^\star$, which are known to be of order $\sqrt{n}$.
• To establish a universal lower bound $\alpha > 1/1.71453$ for the average $\ell^1$-norm of $T^{-1}v_i$ over the cube's vertices, where $T$ is a linear transformation mapping the crosspolytope into the cube.
• To provide computer-generated candidate optimal transformations and distances in dimensions 3 through 8, supporting conjectures on the exact value in low dimensions.
• To explore the minimal value of the average $\ell^1$-norm of inner products between a unit vector and the vertices of the hypercube, which governs the lower bound of the Banach-Mazur distance.

Proposed method

• Derives a lower bound $\alpha$ for the average $\ell^1$-norm of $T^{-1}v_i$ over $v_i \in \{-1,1\}^n$, using a recursive averaging argument on $F_n(x) = \frac{1}{2^n}\sum_{v_i}|\langle x, v_i \rangle|$.
• Applies an induction argument on $n$ to bound $\alpha_n = \min_{\|x\|_2=1} F_n(x)$, showing $\alpha_n \leq \sqrt{2} \prod_{j=4}^{n-1} (1 + \frac{1}{2(j-1)^2}) < 1.71453$, leading to $\alpha > 1/1.71453$.
• Uses numerical optimization via Wolfram Mathematica to compute candidate optimal linear transformations $T$ such that $gC_n^\star \subset C_n$ and $\frac{1}{r}C_n \subset gC_n^\star$, minimizing $r$.
• Employs the identity $d_{BM}(C_n, C_n^\star) = \min_T \max_{v_i \in \{-1,1\}^n} \|T^{-1}v_i\|_1$ under constraints $|x_{ij}| \leq 1$, enabling computer search over $T \in \mathrm{GL}(n,\mathbb{R})$ with bounded entries.
• Analyzes the structure of vertices of the transformed crosspolytope $gC_n^\star$ to identify candidate optimal configurations, especially in low dimensions.
• Conjectures that $\alpha_n = \sqrt{2}$ for all $n$, which would yield a tighter lower bound $\sqrt{n}/\sqrt{2}$, and supports this via numerical evidence in dimensions 4–8.

Experimental results

Research questions

• RQ1 What are the best explicit upper and lower bounds for the Banach-Mazur distance $d_{BM}(C_n, C_n^\star)$, given that it is known to be $\Theta(\sqrt{n})$?
• RQ2 Can the universal constant $\alpha$ in the inequality $\frac{1}{2^n}\sum_{v_i}|\langle x, v_i \rangle| \geq \alpha \|x\|_2$ be improved beyond $\alpha > 1/1.71453$?
• RQ3 What is the exact value of $d_{BM}(C_3, C_3^\star)$, and does the numerical result $9/5$ represent the true minimum?
• RQ4 Can the upper bound $d_{BM}(C_n, C_n^\star) \leq (\sqrt{2}+1)\sqrt{n}$ be improved, and is $\sqrt{n}+3$ a plausible tighter bound?
• RQ5 Are there structural patterns in the optimal linear transformations $T$ that map the crosspolytope into the cube, particularly in low dimensions?

Key findings

• The paper establishes a new lower bound for the Banach-Mazur distance: $d_{BM}(C_n, C_n^\star) \geq \alpha \sqrt{n}$ with $\alpha > 1/1.71453 \approx 0.5833$, derived from the minimal average $\ell^1$-norm of $T^{-1}v_i$ over the cube’s vertices.
• An improved upper bound is proven: $d_{BM}(C_n, C_n^\star) \leq (\sqrt{2}+1)\sqrt{n} \approx 2.414\sqrt{n}$, which improves upon earlier estimates with $C \approx 5.285$.
• In dimension 3, the paper confirms the conjectured value $d_{BM}(C_3, C_3^\star) = \frac{9}{5} = 1.8$, with an explicit optimal transformation matrix provided.
• In dimension 4, the distance is numerically found to be exactly 2, with a symmetric optimal transformation matrix given.
• For dimensions 5 to 8, the paper reports computer-aided numerical results: distances approximately 2.32871, 2.6235, 2.908, and 3.186, respectively, with corresponding transformation matrices.
• The paper conjectures that the true lower bound is $\sqrt{n}/\sqrt{2} \approx 0.7071\sqrt{n}$, and that the upper bound $\sqrt{n}+3$ may be achievable, both of which would significantly improve current bounds.