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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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