From the Mahler Conjecture to Gauss Linking Integrals

By Greg Kuperberg. 2006

202 citations recorded by the source corpus.

Abstract.We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product v(K) = (Vol K)(Vol K°) of the volume of a symmetric convex body $$K \in {\mathbb{R}}^{n}$$ and its polar body K°. The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain $$K^{\diamond} \subseteq K \times K^{\circ}$$ is minimized when K is an ellipsoid. It implies the Mahler conjecture up to a factor of (π/4)nγn, where γn is a monotonic factor that begins at 4/π and converges to $${\sqrt2}$$. This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of cn.The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K◇, with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere Sn-1 and in hyperbolic space Hn-1.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b5f9ac60950319ef19e34e510fe5c1fbed547f6c

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Citations in the corpus

The Hörmander Proof of the Bourgain–Milman Theorem (2012, background): “The best value of c I could get on this way is 4 3 , which is 3 times worse (on the logarithmic scale) than the current record c D 4 due to Kuperberg [7].”
Some functional inverse Santalo inequalities (2008, background): “[11] G. Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom.”
Volume inequalities for asymmetric Wulff shapes (2011, background): “ if K is a simplex with centroid at the origin. It is an important task in convex geometry to ﬁnd sharp lower bounds for the volume of K∗ in terms of other natural geometric quantities of K (see e.g. [8, 10, 19, 34, 36] for results in this direction). Unfortunately, for inequality (6.9), the require”
A volume inequality for polar bodies (2010, background): “What’s especially remarkable about Kuperberg’s inequality is that it provides an explicit value for c.”
New Orlicz affine isoperimetric inequalities (2014, background): “Estimates on the constant c can be found in [19, 33].”
A remark on the Mahler conjecture: local minimality of the unit cube (2009, background): “d for zonoids, [R1], [GMR]. Bourgain and Milman [BM] proved the inequality P(K)1/n > cP(Bn ∞) 1/n, with some constant c > 0 independent of n. The best known constant c = π/4 is due to Kuperberg [Ku]. Note that the exact upper bound for P(K) is known: P(K) 6 P(Bn 2), where Bn 2 is the n-dimensi”
Rogers–Shephard inequality for log-concave functions (2014, background): “The far reaching influence of this inequality becomes evident as it can be found as an ingredient not only in many important works in classical and asymptotic convex geometry [11,23,21,26] but also in many others with a”
Shadow systems of asymmetric Lp zonotopes (2013, unknown): “[1,3,6,7,10,13,18–20,31,33,34,38,39,41]).”
From symplectic measurements to the Mahler conjecture (2013, methodology): “The best known constant nowadays, γ = π/4, is due to Kuperberg [17].”
Dual Orlicz–Brunn–Minkowski theory: Dual Orlicz Lϕ affine and geominimal surface areas (2014, background): “3) where c > 0 is a (universal) constant independent of n and K (see [18, 34] for estimates on c).”