Let __\( \Delta \)__ be an __\( n \)__-dimensional polytope that is simple, that is, exactly __\( n \)__ facets meet at each vertex. An affine function is “mass linear” on __\( \Delta \)__ if its value on the center of mass of __\( \Delta \)__ depends linearly on the positions of the supporting hyperplanes. On the one hand, we show that certain types of symmetries of __\( \Delta \)__ give rise to nonconstant mass linear functions on __\( \Delta \)__. 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 __\( \Delta \)__, then __\( \Delta \)__ is a product of simplices. Our main result is a classification of all smooth polytopes of dimension __\( \leq 3 \)__ 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 __\( (M,\omega,T,\Phi) \)__ with moment polytope __\( \Delta = \Phi(M) \)__. Let
__\[ \mathrm{Symp}_0(M,\omega) \]__
denote the identity component of the group of symplectomorphisms of __\( (M,\omega) \)__. Any linear function __\( H \)__ on __\( \Delta \)__ generates a Hamiltonian __\( \mathbb{R} \)__ action on __\( M \)__ whose closure is a subtorus __\( T_H \)__ of __\( T \)__. We show that if the map
__\[ \pi_1(T_H)\to\pi_1(\mathrm{Symp}_0(M,\omega)) \]__
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
__\[ \pi_1(T)\to\pi_1(\mathrm{Symp}_0(M,\omega)) \]__
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
__\[ \mathrm{Isom}_0(M)\subset\mathrm{Symp}_0(M,\omega) ;\]__
there is a natural compatible complex structure __\( J \)__ on __\( M \)__, and __\( \mathrm{Isom}_0(M) \)__ is the identity component of the group of symplectomorphisms that also preserve this structure. We prove that if the polytope __\( \Delta \)__ supports no essential mass linear functions, then the induced map
__\[ \pi(\mathrm{Isom}_0(M))\to\pi_1(\mathrm{Symp}_0(M,\omega)) \]__
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 __\( \mathbb{C}P^2 \)__ bundle over __\( \mathbb{C}P^1 \)__.