Chebyshev foam (Q1296287)

Let \(E\) be the union of all finite-dimensional Euclidean spaces equipped with the natural norm and real inner product. A closed set \(C\subset E\) is a Chebyshev set if every point in \(E\) has a unique nearest point in \(C\). The construction of a Chebyshev set \(F\subset E\) with the following properties (Chebyshev foam) is presented in this paper: \(E\setminus F\) is dense in \(E\), \(E\setminus F\) has infinitely many components, every component of \(E\setminus F\) is bounded and the infimum of the diameters of the components of \(E\setminus F\) is 0.