Freedman, Michael H. A power law for the distortion of planar sets. (English) Zbl 0642.52006 Discrete Comput. Geom. 2, 345-351 (1987). This interesting paper gives a lower bound for the Lipschitz constants of injective mappings \(f: S_ x\to C_ y\), where \(S_ x\) and \(C_ y\) are finite metric spaces, which may arise when one tries to store a set of planar data in a stack of chips. More precisely, let w be a weight distribution on the edges of the usual cell complex \(K^ n\) associated with the lattice \(Z^ n\) in \(R^ n,\) and define the metric \(d_ w\) on \(Z^ n\) by \(d_ w(p,q)=\inf \{\sum_{e\in P}w(e):\) P is a path in the 1-skeleton of \(K^ n,\) connecting p to \(q\}\). Now we set\(S_ x=Z^ 2\cap [0,x-1]^ 2,\) \(w(e)=1\) for every edge e in \(K^ 2,\) and provide \(S_ x\) with the metric \(d_ w\). We also set \(C_ y=Z^ 3\cap [0,y- 1]^ 3,\) \(v(e)=1\), for every horizontal edge e in \(K^ 3,\) \(v(e)=a>0\), for every verticaledge e between two boundary points of \(C_ y\), \(v(e)=\infty\), otherwise, and provide \(C_ y\) with the metric \(d_ v\). Generalizations to higherdimensional situations, and to other metrics, are indicated. Reviewer: P.Mani MSC: 52A37 Other problems of combinatorial convexity 51F99 Metric geometry Keywords:square; street-metric; cube; foliation-metric; stack of chips PDFBibTeX XMLCite \textit{M. H. Freedman}, Discrete Comput. Geom. 2, 345--351 (1987; Zbl 0642.52006) Full Text: DOI EuDML