Convex Polyhedra Realizing Given Face Areas

TL;DR

This paper proves a necessary and sufficient condition for a set of positive real numbers to be the face areas of a convex polyhedron, specifically that the largest face area does not exceed the sum of the others.

Contribution

It establishes a precise geometric criterion linking face areas to the convex polyhedron realization problem.

Findings

The largest face area is at most the sum of the other face areas.

The condition is both necessary and sufficient for realization.

Provides a complete characterization for face areas of convex polyhedra.

Abstract

Given n >= 4 positive real numbers, we prove in this note that they are the face areas of a convex polyhedron if and only if the largest number is not more than the sum of the others.