Let $T$ be a compact real torus, and $X$ a Hamiltonian $T$-manifold (which you may take to be a smooth complex projective variety) with moment map $\mu:X\rightarrow\frak{t}^\*$. If $\dim(T)=\frac{1}{2}\dim(X)$, then $X$ is classified up to $T$-equivariant symplectomorphism by its moment polytope $\mu(X)$. If we relax this dimension condition, then what information about $X$ might we hope to obtain from an explicit description of its moment polytope? For instance, is the $T$-equivariant cohomology of $X$ nicely related to $\mu(X)$? My question is somewhat vague, so please feel free to answer in any fashion you deem to be acceptable.