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OP-12033OpenGeometry and topology

Mahler volume product constant (C25C_{25})

Mahler volume product constant: determine C25C_{25} or improve its bounds; the repository records π\pi (lower) and 4 (upper).

MSC 52A40 · Inequalities and extremum problems involving convexity in convex geometry

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

Let K⊂RnK\subset\mathbb R^n be a centrally symmetric convex body (compact, convex, with non-empty interior) satisfying K=−KK=-K. Its polar body is

K∘:={y∈Rn: ⟨x,y⟩≤1 for all x∈K}.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 KK is

vp(K):=Voln(K) Voln(K∘).\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! Voln(K) Voln(K∘)=n! vp(K).M(K) := n!\,\mathrm{Vol}_n(K)\,\mathrm{Vol}_n(K^\circ)=n!\,\mathrm{vp}(K).

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

The constant C25C_{25} is the largest cc such that

n! Voln(K) Voln(K∘) ≥ cnfor all n≥1 and all centrally symmetric convex bodies K⊂Rn,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

Voln(K) Voln(K∘) ≥ cnn!.\mathrm{Vol}_n(K)\,\mathrm{Vol}_n(K^\circ)\ \ge\ \frac{c^n}{n!}.

The (symmetric) Mahler conjecture predicts that C25=4C_{25}=4, with extremisers given by Hanner polytopes (in particular, the cube and cross-polytope).

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Background

  • 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≤2n\le 2, and in dimension n=3n=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 (from the repository page)

  • [BM1987] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in Rn\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.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound π\pi; best known upper bound 4. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
44 Trivial For the cube B∞n=[−1,1]nB_\infty^n=[-1,1]^n one has Vol(B∞n)=2n\mathrm{Vol}(B_\infty^n)=2^n and Vol((B∞n)∘)=2n/n!\mathrm{Vol}\big((B_\infty^n)^\circ\big)=2^n/n!, hence M(B∞n)=4nM(B_\infty^n)=4^n and C25≤4C_{25}\le 4. Conjecturally, this is sharp (Mahler conjecture).

Known lower bounds

Bound Reference Comments
c>0c>0 (non-explicit) [BM1987] Bourgain–Milman (reverse Santaló inequality): there exists a universal constant c>0c>0 with M(K)≥cnM(K)\ge c^n for all centrally symmetric convex bodies KK.
π3/16≈1.9379\pi^3/16 \approx 1.9379 [N2012] Nazarov obtained an explicit constant in the symmetric Bourgain–Milman inequality (via a Hörmander/∂ˉ\bar\partial method).
π≈3.1416\pi \approx 3.1416 [K2008] Best known explicit constant to date (Kuperberg).
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Mahler volume product constant ($C_{25}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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