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.