Compactness theorems for geometric packings (Q1604558)

Let \(C\) be a closed subset of \({\mathbb R}^n\) and let \({\mathcal A}\) be any collection of subsets of \({\mathbb R}^n\). Define the topological space \({\mathcal M}({\mathbb R}^n)\) to be the product space \((O(n) \times {\mathbb R}^n)^{\infty}\), where \(O(n)\) denotes the \(n\)-dimensional orthogonal group. The space \({\mathcal M}({\mathbb R}^n)\) parameterizes all possible positionings of the collection \({\mathcal A}\) in \({\mathbb R}^n\). Denote by \({\mathcal P}({\mathcal A},C)\) the set of all packings of \({\mathcal A}\) into \(C\). The author proves that \({\mathcal P}({\mathcal A},C)\) is a closed subset of \({\mathcal M}({\mathbb R}^n)\). This fact yields compactness statements for packings. For example, if for any \(\epsilon>0\) there exists a packing of \({\mathcal A}\) into the homothet \((1+\epsilon)C\), then there exists a packing of \({\mathcal A}\) into \(C\) itself.