Isoperimetric inequalities for infinite hyperplane systems (Q1892436)

Let \(\mathcal S\) be an infinite discrete system of \(k\)-dimensional flats in \(n\)-dimensional Euclidean space. The author considers isoperimetric problems of the following kind: Let \(\mathcal S\) be a hyperplane system of positive density, find sharp upper bounds for the density of the system of \(k\)-flats \((k \in \{0,\dots, n - 2\}\)) that are generated as intersections of hyperplanes in \(\mathcal S\). This problem is considered both for the volume density involving the total \(k\)-dimensional volume of the intersection of the flats in \(\mathcal S\) with a ball and for the number density relative to a convex body \(K\) (involving the number of flats generated by \(\mathcal S\) that meet \(RK\)). The theory of uniform distribution of sequences provides ideas to define a large class of hyperplane systems (uniform systems) for which all necessary densities exist, isoperimetric inequalities can be proved, and systems with extremal intersection densities can be characterized.