In 1992, Bollobás and Meir showed that for every $k \geq 1$ there exists a constant $c_k$ such that, for any $n$ points in the $k$-dimensional unit cube $[0, 1]^k$, one can find a tour $x_1, \dots, x_n$ through these $n$ points with $\sum_{i = 1}^n |x_i - x_{i + 1}|^k \leq c_k$, where $x_{n + 1} = x_1$ and $|x - y|$ is the Euclidean distance between $x$ and $y$. Remarkably, this bound does not depend on $n$, the number of points. They conjectured that the optimal constant is $c_k = 2 \cdot k^{k / 2}$ and showed that it cannot be taken lower than that. This conjecture was recently revised for $k = 3$ by Balogh, Clemen and Dumitrescu, who showed that $c_3 \geq 2^{7/2} > 2 \cdot 3^{3/2}$. It remains open for all $k > 2$, with the best known upper bound $c_k \leq 2.65^k \cdot k^{k / 2} \cdot (1 + o_k(1))$. We significantly narrow the gap between lower and upper bounds on $c_k$, reducing it from exponential to linear. Specifically, we prove that $c_k \leq 2\mathrm{e}(k + 1) \cdot k^{k / 2}$ and $c_k = k^{k / 2} \cdot (2 + o_k(1))$, the latter establishing the conjecture asymptotically. We also obtain analogous results for related problems on Hamiltonian paths, spanning trees and perfect matchings in the unit cube. Our main tool is a new generalization of the ball packing argument used in earlier works.

翻译：1992年，Bollobás和Meir证明了：对任意$k \geq 1$，存在常数$c_k$，使得在$k$维单位立方体$[0, 1]^k$中任意$n$个点，总能找到一条遍历这些点的回路$x_1, \dots, x_n$，满足$\sum_{i=1}^n |x_i - x_{i+1}|^k \leq c_k$（其中$x_{n+1} = x_1$，$|x-y|$为欧氏距离）。值得注意的是，该上界与点数$n$无关。他们推测最优常数为$c_k = 2 \cdot k^{k/2}$，并证明该值不可再降低。近期Balogh、Clemen和Dumitrescu修正了$k=3$的情况，证明$c_3 \geq 2^{7/2} > 2 \cdot 3^{3/2}$。对于所有$k>2$，该问题仍未解决，目前最佳上界为$c_k \leq 2.65^k \cdot k^{k/2} \cdot (1 + o_k(1))$。我们显著缩小了$c_k$下界与上界之间的差距，将其从指数级降至线性级。具体而言，我们证明$c_k \leq 2\mathrm{e}(k+1) \cdot k^{k/2}$以及$c_k = k^{k/2} \cdot (2+o_k(1))$，后者表明该猜想渐近成立。此外，我们还获得了单位立方体中哈密顿路径、生成树和完美匹配等相关问题的类似结果。我们的主要工具是对早期工作中球填充论证的新推广。