One point compactification and Lipschitz normally embedded definable subsets

2023/04/17 by Costa, André, Grandjean, Vincent, Michalska, Maria
#Algebraic Geometry (math.AG) #FOS: Mathematics #Logic (math.LO) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2304.08555

Abstract

A closed subset of ℝ^q, definable in some given o-minimal structure, is Lipschitz normally embedded in ℝ^q if and only if its one-point compactification is Lipschitz normally embedded in the unit sphere \bf S^q( = ℝ^q ∪ \∞ \), i.e. the closure of its image by the inverse of the stereographic projection is Lipschitz normally embedded in \bf S^q. This implies that any closed connected unbounded definable subset of an Euclidean space is definably inner bi-Lipschitz homeomorphic to a Lipschitz normally embedded definable set.

Related

• On smoothness, tangent cones, and the metric geometry of definable sets
• On coarse Lipschitz embeddability into c₀(κ)
• Lipschitz images and dimensions
• Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space
• Local vs. global Lipschitz geometry