Newton–Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

Let I = { I k } k ∈ N I=\{I_k\}k ∈ N be a graded family of monomial ideals. We use the Newton–Okounkov body of I I to: (a) give a characterization for the Noetherian property of the Rees algebra of the family I I ; and (b) present a combinatorial interpretation for the analytic spread of I I . We also apply these results to investigate and give bounds for the generation type and the Veronese degree of the symbolic Rees algebra of a monomial ideal.