A Note on the Boundary Shape of Matrix Polytope Products

Motivated by interval matrix multiplication we consider (matrix) polytopes A ⊆ R, B ⊆ R, m,n, k ∈ N, and investigate the boundary shape of their pointwise product AB := {AB | A ∈ A, B ∈ B}. We prove that AB cannot have outward curved boundary sections while inward curved sections may exist. This is achieved by a simple local extreme point analysis. Results are proved in a more general abstract setting for images of compact sets of (not necessarily finite dimensional) locally convex vector spaces under continuous multilinear mappings. They can be seen as extensions of the Zadeh-Desoer Mapping Theorem which is a fundamental tool in control theory.