We will investigate the combinatorial and metric relationship among several classes of lattice polytopes in order to resolve associated questions motivated by Representation Theory and Algebraic Geometry. Our aim is to explore and analyze the moduli space of toric degenerations of spherical varieties arising from birational sequences and monomial bases of their representations. One guide through the project is a conjecture of Hibi and Li on f-vector dominance of Ehrhart-equivalent polytopes. We are putting this conjecture into a new framework of alcove polytopes or even polytropes, which form the building blocks of tropical convexity.
