Let $M$ be a metric space and let $T$ be a total ordering of its points. For a finite subset $X\subset M$ we calculate the minimal length $l_{opt}(X)$ of a path visiting all its points, the length $l_T(X)$ of the path which visits the points of $X$ with respect to the order $T$, and the ratio $r_T(X) = l_T(X)/l_{opt}(X)$. As J. Bartholdi and J.Platzman noticed in 1982, for the square $[0;1]^2$ and the order corresponding to the self-similar Peano curve all such ratios $r_T(X)$ are bounded by a logarithmic function of $|X|$. For a given metric space the existence of such "good" orders is connected with more traditional properties and invariants, such as hyperbolicity, Assouad-Nagata dimension, number of ends and doubling. The talk is based on joint works with Anna Erschler.
By Jeremie Brieussel
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