What question did this study set out to answer?

Investigate the isoperimetric problem for Voronoi cells of three-dimensional lattices to identify configurations that minimize the scale-invariant isoperimetric quotient.

August 15, 2026Bollettino Della Unione Matematica ItalianaOpen Access

Local minimality of the truncated octahedron for the isoperimetric problem on parallelohedra

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Investigate the isoperimetric problem for Voronoi cells of three-dimensional lattices to identify configurations that minimize the scale-invariant isoperimetric quotient.
Derived an explicit closed formula for the scale-invariant isoperimetric quotient F in terms of six non-negative Selling parameters.
Analyzed the local behavior of F across primary 3D lattice configurations and evaluated an interpolating family connecting body-centered cubic (BCC) and face-centered cubic (FCC) lattices.
Proved that the body-centered cubic lattice (corresponding to the truncated octahedron) is a strict local minimizer of F at fixed volume.
Demonstrated that neither the face-centered cubic lattice nor the simple cubic lattice are local minimizers of F.
Established that the body-centered cubic lattice is the global minimizer of F within the family of lattices interpolating between BCC and FCC.

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Implication

Geometric analysis demonstrates that body-centered cubic lattices strictly minimize the isoperimetric quotient among 3D parallelohedra, highlighting optimal surface-area-to-volume lattice packing.

Cite This Study

Cesaroni et al. (2026) studied this question.

synapsesocial.com/papers/6a80194675c2e31742c855de https://doi.org/10.1007/s40574-026-00542-6

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Implication

Geometric analysis demonstrates that body-centered cubic lattices strictly minimize the isoperimetric quotient among 3D parallelohedra, highlighting optimal surface-area-to-volume lattice packing.

Local minimality of the truncated octahedron for the isoperimetric problem on parallelohedra

Key Points

Discussion

Implication

Cite This Study

Local minimality of the truncated octahedron for the isoperimetric problem on parallelohedra

Key Points

Discussion

Implication

Cite This Study

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