Volume-like property to upper bound lattice points in a convex body

Question

The following question arises in passing in a joint paper that I am working on. Let $K$ be a centrally symmetric convex body in an $n$-dimensional real vector space $V$ which contains a lattice $L$. $L$ yields a natural volume scale fon $V$, and Minkowski established a well-known lower bound on the number of points in $L$ contained in $K$, namely $N \ge (\text{Vol}\ K)/2^n$. You could view this as a primitive estimate, but actually it's often roughly correct, after you take an $n$th root for a fair comparison.

Is there a similar upper bound that (a) is invariant under the $\text{GL}(n,\mathbb{Z})$ stabilizer of $L$, (b) behaves reasonably if you dilate $K$, and (c) can be regarded as useful or simple? Without any further restrictions on $K$, you obviously can't just use its volume, because it could be a thin rod that contains many lattice points of $L$. An extra restriction that holds in our case is that $K$ is a lattice polytope, i.e., the convex hull of finitely many lattice points. What we currently do is ask for basis of $L$, or a lattice that contains $L$, relative to which $K$ is contained in the $\ell^\infty$ ball of radius $c$. Then of course you get $N \le (2c+1)^n$. Or more generally, you could use a lattice parallelepiped. This works in the cases that we need it, but I don't know when it's a good estimate, again with the allowance of an $n$th root. It's also only $\text{GL}(n,\mathbb{Z})$-invariant by fiat.

Answer 1

Oded Regev and I recently published a paper that partially answers this question: https://arxiv.org/abs/1611.05979. In particular, we obtain reasonably tight bounds for the case when $K$ is a Euclidean ball in terms of the quantity $\min_{L' \subseteq L} \det(L')^{1/\mathrm{rank}(L')}$:

Section 9 of Dadush and Regev (https://arxiv.org/abs/1606.06913) has some discussion about the case of general convex bodies (as well as possible strengthenings of the above). In particular, they show that a natural generalization of the above to arbitrary convex bodies fails unless $t \geq \Omega(n^{1/4})$, but it holds for $t = O(n)$.