On the Brunn-Minkowski and Aleksandrov-Fenchel Inequalities
Institution: | Chalmers University of Technology |
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Department: | |
Year: | 2014 |
Record ID: | 1371358 |
Full text PDF: | http://studentarbeten.chalmers.se/publication/199244-on-the-brunn-minkowski-and-aleksandrov-fenchel-inequalities |
The Brunn-Minkowski inequality has a wide range of generalizations and its applications spread throughout many mathematical fields. Using a inequality by Brascamp-Lieb a functional version of Brunn-Minkowski is found in Prekopa's theorem and the Prekopa- Leindler inequality. We demonstrate the wide applicability of the Brunn-Minkowski inequality and its functional counterparts. Using basic properties of differential forms we find an alternate proof of the classical result that the volume of a Minkowksi sum is a polynomial. Further by applying techniques from the realm of differential forms an attempt is made to simplify and generalize the proof of the Aleksandrov-Fenchel inequality.