Abstract
We study a polyhedron with $n$ vertices of fixed volume having the minimum surface area. Completing the proof of Fejes Tóth, we show that all faces of a minimum polyhedron are triangles, and further prove that a minimum polyhedron does not allow deformation of a single vertex. We also present possible minimum shapes for $n ≤ 12$. Some of them are quite unexpected, in particular $n = 8$.
Funding Statement
The author is supported by JSPS Grant-in-aids 17K05159, 17H02849,
BBD30028.
Citation
Shigeki Akiyama. "Minimum polyhedron with $n$ vertices." Hiroshima Math. J. 51 (2) 111 - 137, July 2021. https://doi.org/10.32917/h2018079
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