Titelaufnahme

Titel
Optimal transport and geometric inequalities / Robert Bruno Schlichtner
VerfasserSchlichtner, Robert Bruno
Begutachter / BegutachterinSchuster, Franz
Erschienen2012
Umfang58 Bl.
HochschulschriftWien, Techn. Univ., Dipl.-Arb., 2014
SpracheEnglisch
DokumenttypDiplomarbeit
Schlagwörter (DE)optimaler Transport / isoperimetrische Ungleichung / Sobolev Ungleichung
Schlagwörter (EN)optimal transport / isoperimetric inequality / Sobolev inequality
URNurn:nbn:at:at-ubtuw:1-77164 Persistent Identifier (URN)
Zugriffsbeschränkung
 Das Werk ist frei verfügbar
Dateien
Optimal transport and geometric inequalities [0.5 mb]
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Zusammenfassung (Englisch)

In this thesis we will first introduce some important concepts connected to optimal transport, including the Brenier map, a map from R^n to R^n derived from a convex potential pushing forward one probability measure to another, and the Monge-Ampère equation, a partial differential equation, linking the densities of these measures and the Brenier map. The second and main part presents proofs of several important geometric and analytic inequalities, namely the Brunn-Minkowski inequality, the Prèkopa-Leindler inequality, the Minkowski inequality, the (reverse) Brascamp-Lieb inequality and the Gagliardo-Nirenberg-Sobolev inequality, which are based on the aforementioned tools of optimal transportation.