Mahler volume product constant

Description of constant

Let $K \subset R^{n}$ be a centrally symmetric convex body (compact, convex, with non-empty interior) satisfying $K = - K$ . Its polar body is

$K^{\circ} := {y \in R^{n} : ⟨ x, y ⟩ \leq 1 for all x \in K} .$

The volume product of $K$ is

$vp (K) := Vol_{n} (K) Vol_{n} (K^{\circ}) .$

It is common (and convenient) to absorb the factorial and define the Mahler volume

$M (K) := n! Vol_{n} (K) Vol_{n} (K^{\circ}) = n! vp (K) .$

For centrally symmetric $K$ , this quantity is invariant under invertible linear transformations, so it makes sense to ask for a lower bound of the form $M (K) \geq c^{n}$ .

The constant $C_{25}$ is the largest $c$ such that

$n! Vol_{n} (K) Vol_{n} (K^{\circ}) \geq c^{n} for all n \geq 1 and all centrally symmetric convex bodies K \subset R^{n},$

equivalently

$Vol_{n} (K) Vol_{n} (K^{\circ}) \geq \frac{c ^{n}}{n !} .$

The (symmetric) Mahler conjecture predicts that $C_{25} = 4$ , with extremisers given by Hanner polytopes (in particular, the cube and cross-polytope).

Known upper bounds

Bound	Reference	Comments
$4$	Trivial	For the cube $B_{\infty}^{n} = [- 1, 1]^{n}$ one has $Vol (B_{\infty}^{n}) = 2^{n}$ and $Vol ((B_{\infty}^{n})^{\circ}) = 2^{n} / n!$ , hence $M (B_{\infty}^{n}) = 4^{n}$ and $C_{25} \leq 4$ . Conjecturally, this is sharp (Mahler conjecture).

Known lower bounds

Bound	Reference	Comments
$c > 0$ (non-explicit)	[BM1987]	Bourgain–Milman (reverse Santaló inequality): there exists a universal constant $c > 0$ with $M (K) \geq c^{n}$ for all centrally symmetric convex bodies $K$ .
$π^{3} /16 \approx 1.9379$	[N2012]	Nazarov obtained an explicit constant in the symmetric Bourgain–Milman inequality (via a Hörmander/ $\partial ˉ$ method).
$π \approx 3.1416$	[K2008]	Best known explicit constant to date (Kuperberg).

Additional comments and links

The Blaschke–Santaló inequality gives the opposite extremal problem: for any convex body (after translating to its Santaló point), the volume product is maximized by ellipsoids.
The symmetric Mahler conjecture is known in low dimensions: it is true in dimensions $n \leq 2$ , and in dimension $n = 3$ it was proved by Iriyeh–Shibata.
There is also a non-symmetric Mahler conjecture (minimizers conjectured to be simplices); see the surveys below for background and many partial results (e.g. for unconditional bodies, zonoids, and other symmetry classes).
Surveys: [Mak2015], [FMZ2023].
Wikipedia: https://en.wikipedia.org/wiki/Mahler_volume

References

[BM1987] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in $R^{n}$ , Invent. Math. 88 (1987), 319–340.
[N2012] F. Nazarov, The Hörmander proof of the Bourgain–Milman theorem, in: Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics 2050, Springer, 2012.
[K2008] G. Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal. 18 (2008), no. 3, 870–892.
[IS2020] H. Iriyeh and M. Shibata, Symmetric Mahler's conjecture for the volume product in the 3-dimensional case, Duke Math. J. 169 (2020), no. 6.
[Mak2015] E. Makai Jr., The recent status of the volume product problem, arXiv:1507.01473.
[FMZ2023] M. Fradelizi, M. Meyer, and A. Zvavitch, Volume Product, arXiv:2301.06131.

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.

Current state

The problem