TITLE: Minkowski's theorem for non-0-symmetric sets
QUESTION [7 upvotes]: Let $\Lambda \subseteq \mathbb{R}^n$ be a full-rank lattice, i.e. $\Lambda = A \mathbb{Z}^n$ for some $A \in \mathrm{GL}_n (\mathbb{R})$, and let $C \subseteq \mathbb{R}^n$ be a $0$-symmetric convex body. Then Minkowski's theorem asserts that
$$
\# | \Lambda \cap C | \geq \frac{|C|}{2^n | \mathbb{R}^n / \Lambda|}.
$$
I was curious if there is a version of Minkowski's theorem that holds for convex bodies not necessarily symmetric around $0$. Clearly, the body should contain $0$, otherwise we might have $\# | \Lambda \cap C | = 0$. Are there other conditions on $C$ than symmetry around $0$ that still allow for a lower bound on the number of lattice points in $C$?
Any comment or reference is highly appreciated.

REPLY [3 votes]: I am not sure whether this will necessarily be of interest to you, but Athreya and Margulis jointly proved a probabilistic version of the Minkowski Theorem; here is the arXiv version: https://arxiv.org/pdf/0811.2806.pdf . The random Minkowski theorem is Theorem 2.2 on the third page. Str\"ombergsson then showed that the bound that they obtain is sharp: see https://arxiv.org/pdf/1008.3805.pdf . In a recent preprint, I then managed to generalize the random Minkowski theorem of Athreya-Margulis to higher "probabilistic successive minima" (so to speak); see https://arxiv.org/pdf/1909.05205.pdf . (I should probably change "Lebesgue measurable" to "Borel measurable," though.)