The sausage conjecture holds for convex hulls of moderately bent sausages (Q1297782)

Consider a finite packing of balls of radius \(1\) in \({\mathbf R}^d\). The ``sausage conjecture'' claims that for \(d\geq 5\) the volume of the convex hull of such a configuration is minimized if the centers of the balls lie on a line. Some partial results are known - in particular, this was proved for \(d\geq 42\) by \textit{V. Betke} and \textit{M. Henk} [Discrete Comput. Geometry 19, No. 2, 197-227 (1998; Zbl 0897.52005)]. The paper under review gives a lower bound (for \(d\geq 5\)) for the volume of the convex hull of \(n\) disjoint unit balls, under the condition that there is no acute angle (\(<\pi/2\)) between two balls that are both ``close'' (distance of the centers \(<2\sqrt{2}\)) to a given ball. Under the extra assumption that two balls have only one close neighbor, the lower bound implies the estimate needed by the sausage conjecture. Thus it establishes a stability result for the sausage conjecture.