Cones and convex bodies with modular face lattices

TL;DR

This paper characterizes convex bodies with modular face lattices, showing they are either sections of hermitian matrix cones or have specific dimensions, revealing deep geometric and algebraic structures.

Contribution

It establishes a classification of convex bodies with modular, irreducible face lattices, linking them to hermitian matrix cones or specific dimensions.

Findings

Convex bodies with modular face lattices are homeomorphic to sections of hermitian matrix cones.

Such bodies are either sections of hermitian cones or have dimensions 8, 14, or 26.

The face lattice structure determines the geometric nature of the convex body.

Abstract

If a convex body C has modular and irreducible face lattice (and is not strictly convex), there is a face-preserving homeomorphism from C to a section of a cone of hermitian matrices or C has dimension 8, 14 or 26.