# Minimum polyhedron with $n$ vertices

## TL;DR

This paper investigates the shape of polyhedra with a fixed number of vertices and volume that minimize surface area, proving all faces are triangles and exploring specific minimal configurations for small n.

## Contribution

It completes Fejes Toth's proof by showing all faces are triangles and that such polyhedra are rigid, also identifying potential minimal shapes for n up to 12.

## Key findings

- All faces of minimum polyhedra are triangles
- Minimum polyhedra are rigid with no vertex deformation
- Identified specific minimal shapes for n ≤ 12

## Abstract

We study a polyhedron with $n$ vertices of fixed volume having minimum surface area. Completing the proof of Fejes Toth, 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 $n\le 12$, some of them are quite unexpected, in particular $n=8$.