Log-Lipschitz embeddings of homogeneous sets with sharp logarithmic exponents and slicing the unit cube

2010/07/26 by James C. Robinson, James C Robinson, Robinson, James C
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Dynamical Systems (math.DS) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Metric Geometry (math.MG) #math.AP #math.DS #math.MG

paper · pdf · doi:10.48550/arxiv.1007.4570

arxiv created 2010/07/26 · openalex publication_date 2010/07/26 · arxiv updated 2010/07/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

If X is a subset of a Banach space with X-X homogeneous, then X can be embedded into some \Rⁿ (with n sufficiently large) using a linear map L whose inverse is Lipschitz to within logarithmic corrections. More precisely, c (‖x-y‖)/(| log‖x-y‖ |^α)≤|Lx-Ly|≤ c‖x-y‖ for all x,y∈ X with ‖x-y‖<δ for some δ sufficiently small. A simple argument shows that one must have α>1 in the case of a general Banach space and α>1/2 in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved. While the argument in a general Banach space is relatively straightforward, the Hilbert space case relies on a result due to Ball (Proc. Amer. Math. Soc. 97 (1986) 465-473) which guarantees that the maximum volume of hyperplane slices of the unit cube in \R^d is √2, in dependent of d.

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