The Mahler conjectures, functional inequalities and polar-product symplectic width
The following describes the scope of the Lean formalization related to the following accompanying paper(s):
- The symmetric Mahler conjecture and its equality cases
- The Mahler Conjecture for General Convex Bodies
- Symplectic Balls in Symmetric Polar Products
Scope
The symmetric Mahler conjecture predicts $|K||K^\circ|\ge4^n/n!$ for every origin-symmetric convex body $K\subset\mathbb R^n$. The formalization establishes this for every $n\ge1$ and characterizes equality exactly by invertible linear images of Hanner bodies, built from intervals using Cartesian products and convex-hull joins.
The nonsymmetric Mahler conjecture and the functional inequalities are not included.
The general Mahler conjecture gives a sharp lower bound for the volume product of a convex body and its polar. For every $n\ge1$ and compact convex body $K\subset\mathbb R^n$ with nonempty interior, the formalization proves $\inf_{z\in\mathrm{int},K}|K|,|(K-z)^\circ|\ge (n+1)^{n+1}/(n!)^2$. Equality holds exactly when $K$ is a simplex. The paper's functional inequality is outside this selected statement.
For an origin-symmetric convex body $K\subset\mathbb R^n$, $n\ge2$, the formalized result determines the symplectic ball capacity of $\mathrm{int},K\times\mathrm{int},K^\circ$: its Gromov width is $4$, and every ball of capacity $0<c<4$ embeds symplectically into it. The normalization assigns capacity $\pi r^2$ to a ball of radius $r$.
No boundary smoothness or strict convexity is assumed. An embedding at capacity exactly $4$ is not asserted.
Comparator links
| Result | Comparator statement |
|---|---|
| Symmetric Mahler inequality | MahlerConjecture.lean |
| Symmetric Mahler equality characterization | SymmetricMahlerEquality.lean |
| General Mahler inequality and simplex equality cases | GeneralMahler.lean |
| Gromov width of symmetric polar products | SymmetricPolar.lean |