Squared edge bound for tours in the unit square (${\sum }_i\Vert x_{i+1}-x_i{\Vert }^2\le 4$)

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.

Given finitely many points in a right triangle with hypotenuse endpoints $a,b$, there is a path from $a$ to $b$ visiting them all with sum of squared edge lengths at most $\Vert a-b{\Vert }^2$. Repeated altitude subdivision reduces to cells containing one point. Joining the child paths preserves the squared-cost bound by the Pythagorean theorem; deleting an auxiliary right-angle vertex preserves it by the law of cosines.