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From the Mahler conjecture to Gauss linking integrals. (English) Zbl 1169.52004
Mahler’s conjecture states that the volume product \(v(K)= (\text{Vol}K)(\text{Vol}K^o)\), where \(K\) is a symmetric convex body in \({\mathbb R}^n\) and \(K^o\) is its polar body with respect to the centre, attains its minimum if \(K\) is a parallelepiped (though for \(n>2\) not only in this case). J. Bourgain and V. D. Milman [Invent. Math. 88, 319–340 (1987; Zbl 0617.52006)] have shown that there exists a dimension-independent constant \(c>0\) such that \(v(K) \geq c^n v(C^n)\), where \(C^n\) is an \(n\)-cube.
The present paper gives a new proof of this fact, which is remarkable under several aspects. The obtained constant \(c=4/\pi\) is the best one presently known. The proof uses a version of the Gauss linking integral and establishes a version of the so-called bottleneck conjecture, asserting that the volume of a certain domain \(K^\diamond\subseteq K\times K^o\) is minimized when \(K\) is an ellipsoid.

MSC:
52A40 Inequalities and extremum problems involving convexity in convex geometry
46B07 Local theory of Banach spaces
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