I can't say that what I'll relate is fundamental, but it does fit into the new ideas category. Since I and (my collaborator) Florent Balacheff have given talks on the subject and the paper will be in the ArXiv in a few days I feel free to comment on it. This post is an annoucement of joint work with Florent Balacheff and Kroum Tzanev.
As you comment, the basic result in the geometry of numbers is Minkowski's (first) theorem: If the volume of a $0$-symmetric convex body $K \subset \mathbb{R}^n$ is at least $2^n$, then $K$ contains a non-zero integer point.
But what happens when the body is not $0$-symmetric? It is easy to see that Minkowski's theorem fails completely, but that's because one is not thinking symplectically. By using some Hamiltonian dynamics of the sort Balacheff and I used to study isosystolic inequalities in this paper, we guessed that the "right" result should be the following:
Conjecture. If a convex body in $\mathbb{R}^n$ contains no integer point other than the origin, then the volume of its dual body with respect to the origin is at least (n+1)/n!
In other words, one should have a sort of uncertainty principle: if the origin is localized as the unique integer point inside a convex body, the dual body cannot be too small. In fact, its volume is bounded below by $(n+1)/n!$. Another formulation of the conjecture that seems more elementary goes as follows:
If every hyperplane $m_1x_1 + \cdots m_nx_n = 1$, where the $m_i$ are integers not all equal to zero, intersects a convex body $K \subset \mathbb{R}^n$, then the volume of $K$ is at least $(n+1)/n!$
We proved the conjecture in the case $n = 2$ and the asymptotic version:
Theorem. There exists a (universal) constant $C \leq 1$ such that if a convex body $K \subset \mathbb{R}^n$ contains no integer point other than the origin, then the volume of $K^*$ is at least $C^n(n+1)/n!$.
In fact, this result is equivalent to Bourgain-Milman. Moreover, it easily implies the asymptotic version of a conjecture of Ehrhart:
Theorem. There exists a universal constant $c \geq 1$ such that if $K \subset \mathbb{R}^n$ is a convex body with barycenter at the origin and containing no other integer point, then the volume of $K$ is at most $c^n (n+1)^n/n!$.
However, what is really interesting for us is that at least in the case $n=2$ the result trascends the geometry of numbers and is really a result in Hamiltonian dynamics. I just need a definition:
Definition. A hypersurface in the cotangent bundle of a manifold $M$ is said to be optical if its intersection with every cotangent space is a convex hypersurface enclosing the origin.
To an optical hypersurface in the cotangent of a compact manifold we can associate two numbers: the symplectic volume of the region enclosed by $\Sigma$ and the least action of its periodic characteristics.
Theorem. An optical hypersurface $\Sigma$ in the cotangent space of the two-torus carries a periodic characteristic whose action is less than or equal to the square root of two-thirds the symplectic volume enclosed by $\Sigma$.
The inequality is sharp.
Finsler geometers will be happier if I translate: If the Holmes-Thompson volume of a (non-reversible) Finsler $2$-torus $(T^2,F)$ is $3/2\pi$, then $(T^2,F)$ carries a (non-contractible) periodic geodesic of length at most $1$.
In other words, this is the (non-reversible) Finsler version of Loewner's systolic inequality. The reversible Finsler version (replace $3/2\pi$ by $2/\pi$) is due to Stéphane Sabourau and can be found here.
