Abstract |
The subject of this work is Mahler's conjecture and the inverse Santalό inequality. Mahler's conjecture states that the volume product $|K|\cdot |K^{\circ }|$ of a symmetric convex body and its polar body satisfies the inequality $$|K|\cdot |K^{\circ }|\geq\frac{4^n}{n!},$$ which means that it is minimal when $K$ is a cube. According to Santalό's inequality, one has $$|K|\cdot |K^{\circ }|\leq |B_n|^2.$$ Equivalently, the volume product is maximal if (and only if) $K$ is an ellipsoid. In the second Chapter we describe a proof of Santalό's inequality and the proof of Mahler's conjecture for two special classes of bodies: zonoids and 1-unconditional bodies (these are results of Reisner and Saint-Raymond respectively). The methods one uses are: Steiner symmetrization, inequalities for log-concave functions, Laplace transform. In the third Chapter we give a proof of the inverse Santalό inequality of Bourgain and Milman: there exists an absolute constant c>0 such that $$|K|\cdot |K^{\circ }|\geq\left (\frac{c}{n}\right )^n$$ for every symmetric convex body K in R^n. The proof uses the method of ``isomorphic symmetrization", which is based on Pisier's inequality and on estimates for the covering numbers (Sudakov's inequality and its dual). The same method gives a proof of Milman's inverse Brunn-Minkowski inequality. All the tools of the proof are developped in the first Chapter (K-convexity, $\ell $-norm, Rademacher projection). In the last part of this work, we give a concise proof of Pisier's result on the existence of alpha -regular bodies (alpha >1/2) in every affine class. The proof uses the method of complex interpolation. The existence of alpha -regular bodies gives an alternative proof of the results of the third Chapter (with better estimates in the case of the inverse Brunn-Minkowski inequality).
|