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We use the hypergraph container theory of Balogh--Morris--Samotij andSaxton--Thomason to obtain general rectilinear approximations and volumeestimates for sequences of bodies closed under certain families of projections.We give a number of applications of our results, including a multicolourgeneralisation of a theorem of Hatami, Janson and Szegedy on the entropy ofgraph limits. Finally, we raise a number of questions on geometric and analyticapproaches to containers.

Rectilinear approximation and volume estimates for hereditary bodies via [0,1]-decorated containers