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Ocello, Antonio (2018) Brenier’s Polar Factorization Theorem. [Laurea triennale]

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Abstract

The goal of this work is to reach the comprehension of the proof of a remarkable result: Brenier's Polar Factorization Theorem. This theorem states that, whenever we have a map h from a bounded subset of R^n with positive Lebesgue measure, which satisfies some conditions it canbe decomposed. These conditions are that h needs to be L^2 vector-valued and nondegenerate, in a sense that we will explore here. The representation of h we achieve with this result is the composition of a rearrangement that is the gradiente a convex function, and a measure-preserving map. The main feature of Brenier regarding this problem was reading it as a problem of Optimal transportation, specially using Kantorovich duality in a very smart way, even if his original motivations were completely out of this area. This is the reason why, first, we will dive a little into the world of Optimal transportation reaching real important results as Kantorovich duality and the Knott-Smith optimality criterion. In the end, instead, we will re-emerge from this field, and finally consider Brenier's polar factorization theorem with its proof.

Item Type:Laurea triennale
Corsi di Laurea Triennale:Scuola di Scienze > Matematica
Subjects:Area 01 - Scienze matematiche e informatiche > MAT/05 Analisi matematica
Codice ID:60845
Relatore:Marson, Andrea
Data della tesi:20 July 2018
Biblioteca:Polo di Scienze > Biblioteca di Matematica

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