Isometric approximation property in euclidean spaces

Abstract

We give a necessary and sufficient quantitative geometric condition for a compact setA⊂Rⁿ to have the following property with a givenc≥1: For everyɛ>0 and for every mapf: A→Rⁿ such that$\left| {\left| {fx - fy} \right| - \left| {x - y} \right|} \right| \leqslant \varepsilon for all x,y \in A$ there is an isometryS: A→Rⁿ such that |Sx−fx|≤cɛ for allx∈A.