Leírás
In 1992, Bollobás and Meir showed that for any $n$ points in the $k$-dimensional unit cube $[0,1]^k$, $k\geq 2$, one can always find a tour $x_1,...,x_n$ through these n points with $|x_1 - x_2|^k + |x_2 - x_3|^k + ... + |x_n - x_1|^k \leq c_k \cdot k^{k/2}$, where $|x - y|$ is the Euclidean distance between $x$ and $y$, and $c_k$ is an absolute constant depending only on $k$. Remarkably, this bound does not depend on $n$, the number of points. They further conjectured that the best possible constant for every $k\geq 2$ is $c_k = 2$ and showed that it cannot be taken lower than that. The conjecture is only known to be true for $k = 2$ due to Newman. Recently, Balogh, Clemen and Dumitrescu revised the conjecture for $k = 3$ by showing that $c_3 \geq 4 \cdot (2/3)^{3/2)}> 2.17$. The best currently known bounds are $c_k <= 1.28 \cdot 5.059^k$ and, for large $k$, $c_k \leq 2.65^k \cdot (1 + o_k(1)).$
In this talk, I will show that $c_k \leq min(6(k+1), 2e(k+2))$, reducing the gap between the lower and upper bounds on $c_k$ from exponential to linear. Moreover, I will give a sketch of a proof that the conjecture holds asymptotically, i.e. $c_k = 2 + o_k(1)$. The main tool is a new generalization of the ball packing argument used in previous works.
https://us06web.zoom.us/j/2572885125?omn=84501050120