For five non-overlapping unit disks, the minimum convex hull perimeter is 10+2π, attained exactly by three contact classes, two with perimeter-preserving flexes and one rigid.
Connelly, Rigidity and Energy, Inventiones Mathematicae, Vol
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Intrinsic Geometry of Hard Disk Clusters
For five non-overlapping unit disks, the minimum convex hull perimeter is 10+2π, attained exactly by three contact classes, two with perimeter-preserving flexes and one rigid.