Low $M^{\ast }$-estimates on coordinate subspaces

Abstract

Let $K$ be a symmetric convex body in ${\mathbf{R}}^n$. It is well-known that for every $\theta \in (0,1)$ there exists a subspace $F$ of ${\mathbf{R}}^n$ with $\dim F=[(1-\theta )n]$ such that $${\mathcal P}_F(K)\supseteq \frac{c\sqrt{θ}} {M_K}D_n\cap F,\leqno (\ast )$$ where ${\mathcal{P}}_F$ denotes the orthogonal projection onto $F$. Consider a fixed coordinate system in ${\mathbf{R}}^n$. We study the question whether an analogue of ($\ast$) can be obtained when one is restricted to choose $F$ among the coordinate subspaces ${\mathbf{R}}^{\sigma },\sigma \subseteq \{1,\dots ,n\}$, with $|\sigma |=[(1-\theta )n]$. We prove several ``coordinate versions" of ($\ast$) in terms of the cotype-2 constant, of the volume ratio and other parameters of $K$. The basic source of our estimates is an exact coordinate analogue of ($\ast$) in the ellipsoidal case. Applications to the computation of the number of lattice points inside a convex body are considered throughout the paper.