Speaker: Enrique Alvarado (ISU)
Title: The Maximum Distance Problem and Rectifiability
Abstract: The maximum distance problem (MDP) is a version of the Analyst’s Traveling Salesman Problem (ATSP). Instead of requiring the salesman to travel to all given locations, one only requires the r-neighborhood (for some given r > 0) to pass through all locations, while ignoring any sort of restrictions on where the salesman should start and end. Mathematically, one minimizes the 1-dimensional Hausdorff measure of the image of a curve of finite length whose r-neighborhood covers a given set E.
In this first part of the talk, we will look at how the MDP minimizers behave as r goes to 0, when E is of fractal, and non-fractal nature. In the second part of the talk, we will investigate a very recent result of my collaborators and I, involving the connection between the Beta numbers (a square sum involving best fitting lines at multiple scales) and the lengths of MDP minimizers.