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

We use the hypergraph container theory of Balogh–Morris–Samotij and Saxton–Thomason to obtain general rectilinear approximations volume estimates for sequences bodies closed under certain families projections. give a number applications our results, including multicolour generalisation theorem Hatami, Janson Szegedy on entropy graph limits. Finally, we raise questions geometric analytic approaches containers.