Let $X$ be an $n$-element point set in the $k$-dimensional unit cube $[0,1]^k$ where $k \geq 2$. According to an old result of Bollob\'as and Meir (1992), there exists a cycle (tour) $x_1, x_2, \ldots, x_n$ through the $n$ points, such that $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} \leq c_k$, where $|x-y|$ is the Euclidean distance between $x$ and $y$, and $c_k$ is an absolute constant that depends only on $k$, where $x_{n+1} \equiv x_1$. From the other direction, for every $k \geq 2$ and $n \geq 2$, there exist $n$ points in $[0,1]^k$, such that their shortest tour satisfies $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} = 2^{1/k} \cdot \sqrt{k}$. For the plane, the best constant is $c_2=2$ and this is the only exact value known. Bollob{\'a}s and Meir showed that one can take $c_k = 9 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ for every $k \geq 3$ and conjectured that the best constant is $c_k = 2^{1/k} \cdot \sqrt{k}$, for every $k \geq 2$. Here we significantly improve the upper bound and show that one can take $c_k = 3 \sqrt5 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ or $c_k = 2.91 \sqrt{k} \ (1+o_k(1))$. Our bounds are constructive. We also show that $c_3 \geq 2^{7/6}$, which disproves the conjecture for $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollob\'as--Meir conjecture is proposed.

翻译：设 $X$ 为 $k$ 维单位立方体 $[0,1]^k$（其中 $k \geq 2$）中的 $n$ 元点集。根据 Bollobás 与 Meir (1992) 的经典结论，存在一条经过这 $n$ 个点的回路（游程）$x_1, x_2, \ldots, x_n$，使得 $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} \leq c_k$ 成立，其中 $|x-y|$ 表示 $x$ 与 $y$ 间的欧氏距离，$c_k$ 是仅依赖于 $k$ 的绝对常数，且 $x_{n+1} \equiv x_1$。另一方面，对于任意 $k \geq 2$ 和 $n \geq 2$，总存在 $[0,1]^k$ 中的 $n$ 个点，其最短游程满足 $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} = 2^{1/k} \cdot \sqrt{k}$。对于平面情形，最佳常数 $c_2=2$ 是目前唯一已知的精确值。Bollobás 与 Meir 证明了对于任意 $k \geq 3$ 可令 $c_k = 9 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$，并猜想对所有 $k \geq 2$ 均有最佳常数 $c_k = 2^{1/k} \cdot \sqrt{k}$。本文显著改进了上界，证明了可取 $c_k = 3 \sqrt5 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ 或 $c_k = 2.91 \sqrt{k} \ (1+o_k(1))$，且所给界具有构造性。同时我们证明了 $c_3 \geq 2^{7/6}$，从而否定了 $k=3$ 时的猜想。本文进一步讨论了该问题与匹配问题、功率分配问题及相关算法问题的联系，并对 Bollobás–Meir 猜想提出了修正版本。