Quantitative straightening of distance spheres

2021/07/20 by David, Guy C., Kaczanowski, McKenna, Pinkerton, Dallas
#28A75 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.09634

Abstract

We study "distance spheres": the set of points lying at constant distance from a fixed arbitrary subset K of [0,1]^d. We show that, away from the regions where K is "too dense" and a set of small volume, we can decompose [0,1]^d into a finite number of sets on which the distance spheres can be "straightened" into subsets of parallel (d-1)-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set K.

Related

• Spheres in the curve complex
• On distance measures for well-distributed sets
• Sharp Lipschitz constants for the distance ratio metric
• Distances and Trees in Dense Subsets of ℤ^d
• Open research areas in distance geometry