Okounkov bodies and the K\"ahler geometry of projective manifolds

Given a projective manifold $X$ equipped with an ample line bundle $L$, we show how to embed certain torus-invariant domains $D \subseteq\mathbb{C}^n$ into $X$ so that the Euclidean K\"ahler form on $D$ extends to a K\"ahler form on X lying in the first Chern class of $L$. This is done using Okounkov bodies $\Delta(L)$, and the image of $D$ under the standard moment map will approximate $\Delta(L)$. This means that the volume of $D$ can be made to approximate the K\"ahler volume of $X$ arbitrarily well. As a special case we can let $D$ be an ellipsoid. We also have similar results when $L$ is just big.