= Squared edge bound for tours in the unit square
{title2=$\sum_i\lVert x_{i+1}-x_i\rVert^2\leq4$}
Every finite point set in the unit square admits a cyclic visiting order with $\sum_i\lVert x_{i+1}-x_i\rVert^2\leq4$. 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 $\operatorname{TS}(X)\leq2\sqrt{|X|}$. The tour minimizing ordinary length need not minimize the sum of squared lengths.
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