Weighted mismatch bound for Euclidean tours ($\mathrm{TS}(x)-\mathrm{TS}(y)\le {\sum }_{i:x_i\ne y_i}{a}_i(x)$)

For every configuration $x$ in the unit square there are nonnegative weights $a_i(x)$ such that ${\sum }_i{a}_i(x{)}^2\le 16$ andUse the sums of the two incident edge lengths in a tour from the squared edge bound for tours in the unit square. Runs of missing vertices can be added by the cheaper of two out-and-back detours. This connects Euclidean travelling salesman tours to Talagrand's convex distance inequality.