Polytopes with mass linear functions, part 1

Let Δ be an n-dimensional polytope that is simple, that is, exactly n facets meet at each vertex. An affine function is "mass linear" on Δ if its value on the center of mass of Δ depends linearly on the positions of the supporting hyperplanes. On the one hand, we show that certain types of symmetries of Δ give rise to nonconstant mass linear functions on Δ. On the other hand, we show that most polytopes do not admit any nonconstant mass linear functions. Further, if every affine function is mass linear on Δ, then Δ is a product of simplices. Our main result is a classification of all smooth polytopes of dimension ≤ 3 which admit nonconstant mass linear functions. In particular, there is only one family of smooth three-dimensional polytopes - and no polygons - that admit "essential mass linear functions," that is, mass linear functions that do not arise from the symmetries described above. In part II, we will complete this classification in the four-dimensional case. These results have geometric implications. Fix a symplectic toric manifold (M,ω,T,Φ) with moment polytope Δ = Φ(M). Let denote the identity component of the group of symplectomorphisms of (M,ω). Any linear function H on Δ generates a Hamiltonian action on M whose closure is a subtorus TH of T. We show that if the map has finite image, then H is mass linear. Combining this fact and the claims described above, we prove that in most cases, the induced map is an injection. Moreover, the map does not have finite image unless M is a product of projective spaces. Note also that there is a natural maximal compact connected subgroup; there is a natural compatible complex structure J on M, and is the identity component of the group of symplectomorphisms that also preserve this structure. We prove that if the polytope Δ supports no essential mass linear functions, then the induced map is injective. Therefore, this map is injective for all four-dimensional symplectic toric manifolds and is injective in the six-dimensional case unless M is a bundle over.