The isoperimetric problem for $3$-polytopes with six vertices
K\'aroly J. B\"or\"oczky, \'Agnes Kov\'acs
TL;DR
This paper proves that among all 3-polytopes with at most six vertices and a fixed volume, the regular octahedron has the smallest surface area, establishing an isoperimetric property.
Contribution
It establishes the minimal surface area property of the regular octahedron among 3-polytopes with up to six vertices for a given volume.
Findings
Regular octahedron minimizes surface area among 3-polytopes with up to six vertices.
The result extends isoperimetric inequalities to a specific class of polytopes.
The proof characterizes the octahedron as the optimal shape under these constraints.
Abstract
We prove that the regular octahedron has the minimal surface area among 3-polytopes of given volume and having at most six vertices.
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Taxonomy
TopicsPoint processes and geometric inequalities · Computational Geometry and Mesh Generation · Advanced Theoretical and Applied Studies in Material Sciences and Geometry