Polytopes with mass linear functions, part I
  We analyze mass linear functions $H$ on simple polytopes $\De$, where a mass linear function is an affine function on $\De$ whose value on the center of mass depends linearly on the positions of the supporting hyperplanes. We show that certain types of symmetries of $\De$ give rise to nonconstant mass linear functions on $\De$. These are called inessential; the others are essential. We also show that most polytopes do not admit any nonconstant mass linear functions. Our main result shows that there is only one family of smooth polytopes of dimension $\leq 3$ which admit essential mass linear functions.   These results have geometric implications. Fix a symplectic toric manifold $(M,\om,T,\Phi)$ with moment polytope $\De = \Phi(M)$; let $\Symp(M,\om)$ be its group of symplectomorphisms. Any linear function $H$ on $\De$ generates a Hamiltonian $\R$ action on $M$ whose closure is a subtorus $T_H$ of $T$. We show that if the map $\pi_1(T_H)\to \pi_1(\Symp(M,\om))$ has finite image, then $H$ is mass linear. Therefore, in most cases the induced map $\pi_1(T) \to \pi_1(\Symp(M,\om))$ is an injection. We also show that this map does not have finite image unless $M$ is a product of projective spaces. Moreover, the inessential $H$ correspond to elements in the kernel of the map $\pi_1(T)\to \Isom(M)$, where the Kahler isometry group $\Isom(M)\subset \Symp(M,\om)$ consists of elements that also preserve the natural compatible complex structure on $M$. Therefore if $\De$ supports no nonconstant essential mass linear $H$, the map $\pi_1(\Isom(M))\to pi_1(\Symp(M,\om)$ is injective. 
Dusa McDuff and Susan Tolman