High-dimensional discrete 1-symmetric convex bodies and dimension-free estimates for maximal functions

Abstract: We investigate the behavior of lattice points in 1-symmetric convex bodies--those invariant under both coordinate permutations and sign changes. In this setting we introduce a discrete analogue of the isotropic constant and establish concentration of mass properties for lattice points that parallel classical mass concentration results for convex bodies in isotropic position. These geometric estimates are applied to prove dimension-free $\ell<sup>p$ bounds ($p \ge 2$) for discrete dyadic maximal operators and for maximal functions in the small-scale regime. Furthermore, we demonstrate that coordinate permutation invariance is a necessary condition by constructing 1-unconditional convex bodies in isotropic position for which these dimension-free estimates fail. Our results provide a framework that generalizes several recent works on dimension-free estimates for discrete maximal functions, including those by Bourgain, Mirek, Stein, and Wróbel.