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

TL;DR

This paper develops rectilinear approximation techniques and volume estimates for hereditary bodies using hypergraph container theory, with applications to graph limit entropy and open questions in geometric analysis.

Contribution

It introduces a novel approach combining hypergraph container theory with geometric approximation for hereditary bodies, extending existing results and proposing new questions.

Findings

General rectilinear approximation methods for hereditary bodies

Volume estimates for bodies under projection families

A multicolour extension of a graph limit entropy theorem

Abstract

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