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Rhombic Dodecahedral Conjecture

The rhombic dodecahedral conjecture asserts that a regular rhombic dodecahedron has the least surface area among three-dimensional parallelohedra of a fixed inradius. Cesaroni and Novaga (2026) proved the sharp inequality

S(P)>=12sqrt(2)r(P)^2,

where S(P) is the surface area and r(P) is the inradius. Equality holds iff P is similar to a regular rhombic dodecahedron. This is a fixed-inradius problem, distinct from minimizing surface area at fixed volume.

See also

Inradius, Parallelohedron, Rhombic Dodecahedron, Truncated Octahedron

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References

Cesaroni, A. and Novaga, M. "The Rhombic Dodecahedral Conjecture." 19 Sep 2026. https://arxiv.org/abs/2610.13975.

Cite this as:

Weisstein, Eric W. "Rhombic Dodecahedral Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RhombicDodecahedralConjecture.html

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