Optimal bounds for the analytical traveling salesman problem

We supply two optimal lower bounds for the solution of the analytical traveling salesman problem in terms of Jones' beta-numbers. We prove a linear lower bound for the length of every connecting curve in the general case. The four corners Cantor set shows that the linear bound is attained. For connected compact sets, we prove an exponential lower bound for their length. The estimate is optimal by the work of Bishop and Jones [2]. Finally, we bridge connected and disconnected cases by taking into account multiplicity of orthogonal projections. Applications to other areas of mathematics are briefly discussed. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

We provide two optimal lower bounds for solving the analytical traveling salesman problem using Jones' beta-numbers. We establish a linear lower bound for the length of every connecting curve in the general case, with the four corners Cantor set demonstrating its achievability. For connected compact sets, we prove an exponential lower bound for their length, which is shown to be optimal based on the work of Bishop and Jones. Finally, we bridge the connected and disconnected cases by considering the multiplicity of orthogonal projections, and briefly discuss applications to other areas of mathematics.