Minkowski's theorem for non-0-symmetric sets

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)

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.)