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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 $\left(\sum_{i=1}^n |x_i - x_{i+1}|^k \right)^{1/k} \leqc_k$, where $|x-y|$ is the Euclidean distance between $x$ and $y$, and $c_k$ isan absolute constant that depends only on $k$, where $x_{n+1} \equiv x_1$. Fromthe 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 valueknown. 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 thebest constant is $c_k = 2^{1/k} \cdot \sqrt{k}$, for every $k \geq 2$. Here wesignificantly 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, relatedproblems, including algorithms, are discussed in this context. A slightlyrevised version of the Bollob\'as--Meir conjecture is proposed.

On a Traveling Salesman Problem for Points in the Unit Cube