GMT Session Speaker: Enrique Alvarado

A Lower Bound on the Reach of Flat Norm Minimizers

Abstract: One measure of the regularity of a curve in R^2 is called the reach of the curve and is equal to thesupremum of the radii of the disks that can be rolled around the curve, always touching the curve in exactly one spot. Curves with corners have reach = 0 since every ball, no matter how small touches the curve in at least two places. And it is not enough that a curve be C^1 for it to have positive reach — positive reach implies a curve is at least C^{1,1}.

In this talk, I will show that the minimizers of the flat norm have positive reach not too much smaller than 1/\lambda where \lambda is the bound on the curvature of the minimizer that is “hard-coded” into the flat norm metric. The proof boils down to a simple comparison argument and some calculations, but, as in most everything in geometric measure theory, there are some details!

Coauthor: Kevin R. Vixie

Biographical Information: Enrique started his career at Gonzaga University where he obtained his undergraduate degree in mathematics in 2014. He is currently a student of Kevin Vixie’s at WSU-Pullman and a member of the Analysis+Data group.