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 $n = 8$.
The author is supported by JSPS Grant-in-aids 17K05159, 17H02849,
"Minimum polyhedron with $n$ vertices." Hiroshima Math. J. 51 (2) 111 - 137, July 2021. https://doi.org/10.32917/h2018079