 Squared edge bound for tours in the unit square

Every finite point set in the unit square admits a cyclic visiting order with ${\sum }_i\Vert x_{i+1}-x_i{\Vert }^2\le 4$. Split the square into two right triangles, apply the quadratic path bound in a right triangle, and remove auxiliary corners using the law of cosines. The Cauchy-Schwarz inequality then gives $\mathrm{TS}(X)\le 2\sqrt{|X|}$. The tour minimizing ordinary length need not minimize the sum of squared lengths.