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July 2021 Minimum polyhedron with $n$ vertices
Shigeki Akiyama
Hiroshima Math. J. 51(2): 111-137 (July 2021). DOI: 10.32917/h2018079

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

Download Citation

Shigeki Akiyama. "Minimum polyhedron with $n$ vertices." Hiroshima Math. J. 51 (2) 111 - 137, July 2021. https://doi.org/10.32917/h2018079

Information

Received: 21 October 2018; Revised: 28 January 2021; Published: July 2021
First available in Project Euclid: 12 July 2021

Digital Object Identifier: 10.32917/h2018079

Subjects:
Primary: 52B55, 52B60

Rights: Copyright © 2021 Hiroshima University, Mathematics Program

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Vol.51 • No. 2 • July 2021
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