Minkowski's theorem for non-0-symmetric sets

Question

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.

Answer 1

The best reference I know on this question, restricted to convex polytopes whose vertices are lattice points (symmetry not assumed), is: Douglas Hensley, Lattice vertex polytopes with interior lattice points, Pacific Journal of Mathematics, Vol 101, No. 1, p. 183-191; MR0688412.

Author's Abstract. Consider a convex polytope with lattice vertices and at least one interior lattice point. We prove that the number of boundary lattice points is bounded above by a function of the dimension and the number of interior lattice points. This extends to arbitrary dimension a result of Scott for the two dimensional case.

and

An excerpt from the review text in MR: The ingenious and elegant proof uses simultaneous Diophantine approximations and some convexity arguments. Especially for k=1, where the results are direct analogues to Minkowski's fundamental theorem on 0-symmetric convex bodies. (Reviewer: J.M. Wills)

Answer 2

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.)

Answer 3

Gruber/Lekkerkerker book "Geometry of Numbers" has a chapter on Minkowski theorem generalizations, and the generalization in case of asymmetric bodies is as follows:

Let $H$ be a bounded convex body containing o as an inner point. Let $\sigma$ be its coefficient of asymmetry and suppose that $V(H) > (1 +\sigma)^n$. Then H contains a lattice point $\neq 0$.

Asymmetry coefficient is $\sigma(H) = \max\limits_{x \in \mathbb{R}} \frac{|x'|}{|x|}$ where $x',x \in H$ and are on one line with origin point.

Proof reference is Mahler K., Ein Ubertragungsprinzip fur konvexe Korper, casopis Pest. Mat. Fyz. 68,93- 102 (1939).