Post-graduate theses
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| Identifier |
uch.math.msc//2001tzibragos |
| Title |
Η εικασία του Mahler και η αντίστροφη ανισότητα Santalό |
| Alternative Title |
Mahler's conjecture and the inverse Santalό inequality |
| Creator |
Tzibragos, George
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| 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).
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| Issue date |
2001-11-01 |
| Date available |
2002-06-10 |
| Collection
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School/Department--School of Sciences and Engineering--Department of Mathematics--Post-graduate theses
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Type of Work--Post-graduate theses
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| Permanent Link |
https://elocus.lib.uoc.gr//dlib/e/2/c/metadata-dlib-2001tzibragos.tkl
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| Views |
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