Equidistribution in All Dimensions of Worst-Case Point Sets for the TSP

Given a set S of n points in the unit square [0, 1]d , an optimal traveling salesman tour of S is a tour of S that is of minimum length. A worst-case point set for the Traveling Salesman Problem in the unit square is a point set S(n) whose optimal traveling salesman tour achieves the maximum possible length among all point sets S ⊂ [0, 1]d , where |S| = n. An open problem is to determine the structure of S(n) . We show that for any rectangular parallelepiped R contained in [0, 1]d , the number of points in S(n) ∩ R is asymptotic to n times the volume of R. Analogous results are proved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree. These equidistribution theorems are the first results concerning the structure of worst-case point sets like S(n).