Low $M^*$-estimates on coordinate subspaces

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 ${\rm dim}F= [(1-\theta )n]$ such that $${\mathcal P}_F(K)\supseteq \frac{c\sqrt{\theta }} {M_K}D_n\cap F,\leqno (\ast )$$ 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 ${\mathbf R}^{\sigma },\; \sigma\subseteq\{1,\ldots,n\}$, 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.