On the equality case of the Alexandrov-Fenchel inequality

Abstract

Denne oppgaven omhandler Alexandrov-Fenchel ulikheten og dens likhets tilfelle. Vi viser til relevante resultater om Hodge-teori, $T$-Hodge-teori og rundt Delzant polytoper, varieteter og linjebunter. Vi avslutter oppgaven med vårt hovedresultat. Det sier at to Delzant polytoper med samme Delzant varietet, gir Alexandrov-Fenchel likhet hvis og bare hvis de er homotetiske.

In this thesis, we look at the Alexandrov-Fenchel inequality and its equality case. We lay out some relevant results in Hodge theory, $T$-Hodge theory, and the theory of Delzant polytopes and their associated varieties and line bundles. We conclude by proving our main result stating that two Delzant polytopes, with the same Delzant variety, attain Alexandrov-Fenchel equality if and only if they are homothetic.