A finite set \(C\) in the \(d\)-dimensional Euclidean space \({\mathbf E}^d\) is called a finite sphere packing if for every \(x,y \in C\) we have \(| x-y| \geq 2\). The density of \(C\) is defined by \[ \delta(C)=\frac{| C| k_d}{V( \text{conv}(C)+B^d)}, \] where \(B^d\) is the \(d\)-dimensional unit ball, \(V(P)\) is the volume of \(P\), and \(k_d=V(B^d)\). The maximal density of sphere packings of \(n\) spheres in \({\mathbf E}^d\) is \[ \delta(d,n)=\max\{ \delta(C) : C \subset {\mathbf E}^d \text{ is packing,} | C| =n\}. \] The packing \(S_n^d=\{ 2iu: u \in S_n^{d-1}\), \(i=1,\ldots,n \}\) is called a sausage arrangement. The longstanding sausage conjecture states that \(\delta(d,n)=\delta(S_n^d)\) for all positive integers \(n\) and \(d \geq 5\). In this very interesting paper the authors continue to show the convenience of the concepts of the parametric density \[ \delta_{\rho}(C)=\frac{| C| k_d}{V( \text{conv}(C)+\rho B^d)}, \] where \(\rho>0\), and the corresponding maximal parametric density \[ \delta_{\rho}(d,n)=\max\{ \delta_{\rho}(C) : C \subset {\mathbf E}^d \text{ is packing, } | C| =n\}. \] The methods, developed by \textit{U. Betke, M. Henk} and \textit{J. M. Wills} [J. Reine Angew. Math. 453, 165--191 (1997; Zbl 0797.52010) and Discrete Comput. Geom. 13, No. 3-4, 297--311 (1995; Zbl 0829.52010)] are optimized here. Moreover, new ideas (such as some measures of the size of a packing) are introduced and developed to result in proving the sausage conjecture in all dimensions \(d \geq 42\) (the previous proof was for \(d \geq 13387\)).