Mathematics Colloquia and Seminars

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We will introduce two main theorems regarding the existence of almost Euclidean sections of high dimensional convex bodies -- Dvoretzky's theorem and the volume ratio theorem. For an $n$-dimensional convex body $K$, the first theorem guarantees the existence of such sections that are arbitrarily close to the Euclidean ball in the Banach-Mazur distance and have dimension not too small (smallness depending on how close to Euclidean one requires). The volume ratio theorem gives the existence(in fact, this existence is with exponentially high probability) of a section of any dimension $1\le k