Let $K \subset {\bf R}^d$ be a symmetric convex body (an open bounded convex neighbourhood of the origin with $K = -K$) with the property that $K + {\bf Z}^d \neq {\bf R}^d$, i.e. the projection of $K$ to the standard torus ${\bf R}^d/{\bf Z}^d$ is not surjective, or equivalently $K$ is disjoint from some coset $x + {\bf Z}^d$ of the standard lattice. My question is: what does this say about the polar body

$$K^* := \{ \xi \in {\bf R}^d: \xi \cdot x < 1 \hbox{ for all } x \in K \}?$$

Intuitively, the property $K + {\bf Z}^d \neq {\bf R}^d$ is a "smallness" condition on K, and is thus a "largeness" condition on $K^*$.

If $K^*$ contains a non-trivial element $n$ of $2 {\bf Z}^d$, then $K$ is contained in the strip $\{ x: |n \cdot x| < 1/2 \}$, and will thus avoid the coset $x+{\bf Z}^d$ whenever $x \cdot n = 1/2$. So this is a sufficient condition for $K + {\bf Z}^d \neq {\bf R}^d$, but it is not necessary. Indeed, if one takes $K$ to be the octahedron

$$K := \{ (x_1,\ldots,x_d) \in {\bf R}^d: |x_1|+\ldots+|x_d| < d/2 \}$$

then $K$ avoids $(1/2,\ldots,1/2)+{\bf Z}^d$, but the dual body

$$ K^* = \{ (\xi_1,\ldots,\xi_d) \in {\bf R}^d: |\xi_1|,\ldots,|\xi_d| < 2/d \}$$

is quite far from reaching a non-trivial element of $2 {\bf Z}^d$.

On the other hand, by using the theory of Mahler bases or Fourier analysis one can show that if $K + {\bf Z}^d \neq {\bf R}^d$, then $K^*$ must contain a non-trivial element of $\varepsilon_d {\bf Z}^d$ for some $\varepsilon_d > 0$ depending only on $d$. However the bounds I can get here are exponentially poor in $d$.

Based on the octahedron example (which intuitively seems to be the "biggest" convex set that still avoids a coset of ${\bf Z}^d$), one might tentatively conjecture that if $K + {\bf Z}^d \neq {\bf R}^d$, then the closure of $K^*$ contains a non-trivial element of $\frac{2}{d} {\bf Z}^d$. I do not know how to prove or disprove this conjecture (though I think the $d=2$ case might be worked out by *ad hoc* methods, and the $d=1$ case is trivial), so I am posing it here as a question.