  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.
