Math Conjectures

Mahler Volume-Product Conjecture

OPENLandmarkConjectureProposed 1939 · Standard version

Canonical statement

For a convex body K⊂RdK\subset\mathbb R^d containing 00 in its interior, put K∘={y:⟨x,y⟩≤1 ∀x∈K}K^\circ=\{y:\langle x,y\rangle\leq1\ \forall x\in K\}. If K=−KK=-K, then
vol⁡(K)vol⁡(K∘)≥4dd!. \operatorname{vol}(K)\operatorname{vol}(K^\circ) \geq\frac{4^d}{d!}.
For arbitrary KK, let s(K)s(K) be its Santaló point, the minimizer of vol⁡((K−x)∘)\operatorname{vol}((K-x)^\circ) over x∈int⁡Kx\in\operatorname{int}K. Then
vol⁡(K)vol⁡((K−s(K))∘)≥(d+1)d+1(d!)2. \operatorname{vol}(K) \operatorname{vol}((K-s(K))^\circ) \geq\frac{(d+1)^{d+1}}{(d!)^2}.
View source LaTeX
For a convex body \(K\subset\mathbb R^d\) containing
\(0\) in its interior, put
\(K^\circ=\{y:\langle x,y\rangle\leq1\ \forall x\in K\}\).
If \(K=-K\), then
\[
  \operatorname{vol}(K)\operatorname{vol}(K^\circ)
    \geq\frac{4^d}{d!}.
\]
For arbitrary \(K\), let \(s(K)\) be its Santaló point, the minimizer of
\(\operatorname{vol}((K-x)^\circ)\) over
\(x\in\operatorname{int}K\). Then
\[
  \operatorname{vol}(K)
  \operatorname{vol}((K-s(K))^\circ)
    \geq\frac{(d+1)^{d+1}}{(d!)^2}.
\]
DG-007:[Mahler1939][BourgainMilman1987][FradeliziMeyerZvavitch2023]
Discrete geometryGeometry and topology

Notes

Mahler asked in 1939 for the minimum of the volume product vol⁡(K)vol⁡(K∘)\operatorname{vol}(K)\operatorname{vol}(K^\circ) of a convex body and its polar, beginning with convex polygons in the plane [Mahler1939]. The conjecture has two standard forms with different extremizers: for origin-symmetric bodies the minimum should be 4d/d!4^d/d!, attained by the cube and its polar the cross-polytope, while for arbitrary bodies, with the polar taken at the Santaló point, the conjectured minimum (d+1)d+1/(d!)2(d+1)^{d+1}/(d!)^2 would be attained by the simplex.

The sharp bounds are known in dimension 22 and for various special classes of bodies. In general dimension the landmark Bourgain–Milman inequality gives a lower bound of the conjectured exponential order, that is, within a factor cdc^d of the conjecture for an absolute constant c>0c>0 [BourgainMilman1987]. The state of the art, together with the many related inequalities, is surveyed by Fradelizi, Meyer, and Zvavitch [FradeliziMeyerZvavitch2023].

In arbitrary dimension both sharp inequalities remain open; the cube–cross-polytope pair and the simplex remain the conjectured extremizers, and establishing the corresponding sharp lower bounds is the outstanding problem.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.