On a Traveling Salesman Problem for Points in the Unit Cube

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) $x1, x2, \ldots, xn$ through the $n$ points, such that $\left(\sum{i=1}ⁿ |xi - x{i+1}|^k \right)^{1/k} \leq ck$, where $|x-y|$ is the Euclidean distance between $x$ and $y$, and $ck$ is an absolute constant that depends only on $k$, where $x{n+1} \equiv x1$. 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 $\left(\sum{i=1}ⁿ |xi - x{i+1}|^k \right)^{1/k} = 2^{1/k} \cdot \sqrt{k}$. For the plane, the best constant is $c2=2$ and this is the only exact value known. Bollob{\'a}s and Meir showed that one can take $ck = 9 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ for every $k \geq 3$ and conjectured that the best constant is $ck = 2^{1/k} \cdot \sqrt{k}$, for every $k \geq 2$. Here we significantly improve the upper bound and show that one can take $ck = 3 \sqrt5 \left(\frac23 \right)^{1/k} \cdot \sqrt{k}$ or $ck = 2.91 \sqrt{k} \ (1+ok(1))$. Our bounds are constructive. We also show that $c3 \geq 2^{7/6}$, 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.