Math 2.0

Convex and discrete geometry Convex geometric analysis and geometric inequalities Conjecture

Mahler conjecture

Builds on Convex bodyLebesgue measureDot product

Statement

Let n≥1n \ge 1 and let K⊆ℝnK \subseteq \R^n be a compact convex body with nonempty interior and K=−KK = -K. Let K∘={y∈ℝn:x⋅y≤1 for all x∈K}K^\circ = \set{y \in \R^n : x \cdot y \le 1 \text{ for all } x \in K} be its polar body, with x⋅yx \cdot y the dot product, and let vol⁡\operatorname{vol} be Lebesgue measure. The conjecture states that vol⁡(K)vol⁡(K∘)≥4nn!,\operatorname{vol}(K) \operatorname{vol}(K^\circ) \ge \frac{4^n}{n!}, which is the value for the cube [−1,1]n[-1, 1]^n.

Remarks

For the cube, vol⁡([−1,1]n)=2n\operatorname{vol}([-1,1]^n) = 2^n and its polar is the cross-polytope of volume 2n/n!2^n / n!, which gives 4n/n!4^n / n!. The conjecture is proved for n=2n = 2, by Mahler, and for n=3n = 3, by Iriyama and Shibata. It is known for some classes of bodies, such as unconditional bodies and zonoids. In general the Bourgain–Milman inequality gives vol⁡(K)vol⁡(K∘)≥cn4nn!\operatorname{vol}(K) \operatorname{vol}(K^\circ) \ge c^n \frac{4^n}{n!} for an absolute constant c>0c > 0, and Kuperberg gave an explicit such constant. The upper bound is the Blaschke–Santaló inequality. There is also a version for convex bodies that are not symmetric, in which the simplex is the conjectured minimizer.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptConvex body, concept, no evidence levelConvex bodyconceptDot product, concept, no evidence levelDot productconceptEuclidean space, object, no evidence levelEuclidean spaceobjectOuter measure, concept, no evidence levelOuter measureconceptLebesgue measure, object, no evidence levelLebesgue measureobjectMahler conjecture, conjecture, no evidence levelMahler conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Convex body concept
  3. Dot product concept
  4. Euclidean space object
  5. Outer measure concept
  6. Lebesgue measure object
  7. Mahler conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Mahler_volume
About this page
TagFFZHNFJCZS
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Kindconjecture
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MSC52A40, 52A20
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AboutConvex body
UsesConvex body, Lebesgue measure, Dot product

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