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Let $\mathcal{I} = \{I_k\}_{k \in \mathbb{N}}$ be a graded family of monomialideal. We use the Newton-Okounkov body of $\mathcal{I}$ to: (a) give acharacterization for the Noetherian property of the Rees algebra of the family;and (b) present a combinatorial interpretation for the analytic spread of thefamily. We also apply these results to investigate the generation type and theVeronese degree of the symbolic Rees algebra of a monomial ideal.

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