Banach uzaylarının geometrisi üzerine bazı sonuçlar

The goal of this study is to get some geometric conclusions from Banach spaces and find some useful applications. In this study, we start by giving some definitions about convexity and measure theory. The Brunn-Minkowski theorem, one of the most significant in convex geometry and functional analysis, is next explored, along with one of its applications, isoperimetric inequality. Second, we prove F. John's theorem, which describes ellipsoids of maximal volume in symmetric convex bodies. The proof which is given is a relatively recent one due to Schuster & Gruber (2005). Finally, we give some basic results in Banach Space theory which are related somehow to the geometric ideas given in John's theorem by using the ideas in Dvoretzky & Rogers (1950).