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Packings, sausages and catastrophes. (English) Zbl 1462.52030

This is a survey on results around the parametric density, which was introduced in 1994 to bridge the gap between finite and infinite packings. Let \(K\) be a convex body in Euclidean space \({\mathbb R}^d\). A packing set of \(K\) is a set \(C\subset{\mathbb R}^d\) satisfying \(\mathrm{int}(K+x)\cap\mathrm{int}(K+y)=\emptyset\) for any distinct \(x,y\in C\). For a finite packing set \(C\) and for \(\rho>0\), the parametric density of \(C\) with respect to \(K\) and \(\rho\) is defined by \[ \delta_\rho(K,C)=\frac{\# C\,\mathrm{vol}(K)}{\mathrm{vol}(\mathrm{conv}\, C +\rho K)}.\] Of interest is then \(\delta_p(K,n)=\sup \delta_\rho(K,C)\), where the supremum is over all \(n\)-element packing sets of \(K\), and also \(\delta_\rho(K)=\limsup_{n\to\infty} \delta_\rho(K,n)\). The paper surveys relations between the ordinary packing density and these parameters, and further ones derived from them, and lists several known estimates. Questions about sausage configurations are emphasized. Also the special roles of balls and of lattice packings are illuminated.

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
11H31 Lattice packing and covering (number-theoretic aspects)
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