Mahler volume product constant
Description of constant
Let $K\subset\mathbb R^n$ be a centrally symmetric convex body (compact, convex, with non-empty interior) satisfying $K=-K$. Its polar body is
\[K^\circ := \left\{y\in\mathbb R^n:\ \langle x,y\rangle \le 1\ \text{for all }x\in K\right\}.\]The volume product of $K$ is
\[\mathrm{vp}(K) := \mathrm{Vol}_n(K)\,\mathrm{Vol}_n(K^\circ).\]It is common (and convenient) to absorb the factorial and define the Mahler volume
\[M(K) := n!\,\mathrm{Vol}_n(K)\,\mathrm{Vol}_n(K^\circ)=n!\,\mathrm{vp}(K).\]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 $M(K)\ge c^n$.
The constant $C_{25}$ is the largest $c$ such that
\[n!\,\mathrm{Vol}_n(K)\,\mathrm{Vol}_n(K^\circ)\ \ge\ c^n \quad\text{for all }n\ge 1\text{ and all centrally symmetric convex bodies }K\subset\mathbb R^n,\]equivalently
\[\mathrm{Vol}_n(K)\,\mathrm{Vol}_n(K^\circ)\ \ge\ \frac{c^n}{n!}.\]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 $\mathrm{Vol}(B_\infty^n)=2^n$ and $\mathrm{Vol}\big((B_\infty^n)^\circ\big)=2^n/n!$, hence $M(B_\infty^n)=4^n$ and $C_{25}\le 4$. 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 $M(K)\ge c^n$ for all centrally symmetric convex bodies $K$. |
| $\pi^3/16 \approx 1.9379$ | [N2012] | Nazarov obtained an explicit constant in the symmetric Bourgain–Milman inequality (via a Hörmander/$\bar\partial$ method). |
| $\pi \approx 3.1416$ | [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 $n\le 2$, 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 $\mathbb R^n$, 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.
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.