Polytopes with Mass Linear Functions, Part I

We analyze mass linear functions on simple polytopes ∆, where a mass linear function is an affine function on ∆ whose value on the center of mass depends linearly on the positions of the supporting hyperplanes. On the one hand, we show that certain types of symmetries of ∆ give rise to nonconstant mass linear functions on ∆. We call mass linear functions which arise in this way inessential; the others are called essential. 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 ∆, then ∆ is affine equivalent to a product of simplices. Our main result is a classification of all 2-dimensional simple polytopes and 3-dimensional smooth polytopes which admit nonconstant mass linear functions. In particular there is only one family of smooth polytopes of dimension ≤ 3 which admit essential mass linear functions. In part II, we will complete this classification in the 4dimensional case. These results have geometric implications. Fix a symplectic toric manifold (M,ω, T,Φ) with moment polytope ∆ = Φ(M). Let Symp0(M,ω) denote the identity component of the group of symplectomorphisms of (M,ω). Any linear function H on ∆ generates a Hamiltonian R action onM whose closure is a subtorus TH of T . We show that if the map π1(TH) → π1(Symp0(M,ω)) has finite image, then H is mass linear. By the claims described above, this implies that in most cases the induced map π1(T )→ π1(Symp0(M,ω)) 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 Isom0(M) ⊂ Symp0(M,ω); there is a natural compatible complex structure J on M , and Isom0(M) is the identity component of the group of symplectomorphisms that also preserve this structure. We prove that if the polytope ∆ supports no nontrivial essential mass linear functions, then the induced map π1(Isom0(M)) → π1(Symp0(M,ω)) is injective. Therefore, this map is injective for all 4-dimensional symplectic toric manifolds. It is also injective in the 6-dimensional case unless M is a CP 2 bundle over CP . with Appendix written with V. Timorin