Pome5C4A19C5ED0BLemma 19.7.7.5C4A19C5ED0B Let $K\in \mathcal{K}_ n$ be a convex body with positive volume and $K'\in \mathcal{K}_ n$. Assume that, for some $k\in \mathbb Z_{>0}$, $K'$ contains $K\cap (k^{-1}\mathbb Z)^ n$. Then
Proof. Let $x\in K_{n^{1/2}k^{-1}}$. By assumption, the closed ball $B$ with center $x$ and radius $n^{1/2}k^{-1}$ is contained in $K$. Observe that $x$ can be written as a convex combination of points in $B\cap (k^{-1}\mathbb {Z})^ n$, which are contained in $K'$ by assumption. It follows that $x\in K'$. $\square$
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