Barker--Larman 问题在维度 $4$ 中的一个近似反例
An approximate counterexample to the Barker--Larman problem in dimension $4$
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中文总结 AI 辅助
本文构造了维度4中一族凸体，其切于内接球的截面面积近似恒定，但内外半径差显著，从而为Barker--Larman问题提供了近似反例。
中文摘要 AI 辅助
Barker--Larman 问题询问：若一个凸体 $K \subseteq \mathbb R^n$ 包含欧几里得球 $\mathbb B_n$，且所有与 $\mathbb B_n$ 相切的超平面截 $K$ 所得的 $(n-1)$ 维体积为常数，则该凸体是否必定为欧几里得球？本文证明了指向维度 $4$ 中否定答案的一个结果。取 $\lambda_0 = 4 \sqrt{3}\pi$ 及任意 $N \in \mathbb N$，我们得到一族凸体 $K_{\lambda,N}$，其中 $\lambda \in (\lambda_0-r_N, \lambda_0 + r_N)$，使得 $K_{\lambda,N}$ 被与欧几里得球相切的超平面截得的面积与 $\lambda$ 的差在 $c |\lambda - \lambda_0|^{\frac{N+1}2}$ 之内，而 $K_{\lambda,N}$ 的外半径与内半径之差大于 $C |\lambda - \lambda_0|$。这些凸体 $K_{\lambda,N}$ 通过径向函数构造为 \\[\rho_{K_{\lambda,N}}(t) = \cos\left( \sum_{n=0}^N \frac{(\lambda-\lambda_0)^n}{n!} \varphi_n(t) \right)^{-1},\\] 其中 $t \in [0,2\pi), \lambda \in \mathbb R$，且 $\varphi_n$ 是可由显式计算的三角多项式。当 $N \to \infty$ 时内幂级数的收敛性（此问题尚未解决）将蕴含维度 $4$ 中 Barker--Larman 问题的否定答案。作为例子，我们得到一个凸体，其外半径与内半径之差大于 $0.176$，而截面面积振荡小于 $3 \times 10^{-7}$。
英文摘要
The Barker--Larman problem asks if a convex body $K \subseteq \mathbb R^n$ containing the Euclidean ball $\mathbb B_n$, such that all the sections of $K$ by hyperplanes tangent to $\mathbb B_n$ have constant $(n-1)$-dimensional volume, must necessarily be a Euclidean ball. In this paper we show a result pointing to a negative answer in dimension $4$. Taking $λ_0 = 4 \sqrt{3}π$ and any $N \in \mathbb N$, we obtain the existence of a family of convex bodies $K_{λ,N}$ with $λ\in (λ_0-r_N, λ_0 + r_N)$, such that the sections of $K_{λ,N}$ by hyperplanes tangent to the Euclidean ball, have area within $c |λ- λ_0|^{N+1}$ of $λ$, while the difference between outradius and inradius of $K_{λ,N}$ is larger than $C |λ- λ_0|$. The bodies $K_{λ,N}$ are constructed via radial functions as \[ρ_{K_{λ,N}}(t) = \cos\left( \sum_{n=0}^N \frac{(λ-λ_0)^n}{n!} φ_n(t) \right)^{-1},\] where $t \in [0,2π), λ\in \mathbb R$ and $φ_n$ are trigonometric polynomials that can be computed explicitly. The convergence of the inner power series when $N \to \infty$ (which is left open) would imply a negative answer to the Barker--Larman problem in dimension $4$. As an example we obtain a convex body whose outradius and inradius differ by more than $0.176$, and the area of the sections oscillate by less than $3 \times 10^{-7}$.
