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The concept of symmetry breaking and the emergence of corresponding local order parameters constitute the pillars of modern day many body physics. I will demonstrate that the existence of symmetry breaking is a consequence of the geometric structure of the convex set of reduced density matrices of all possible many body wavefunctions. The surfaces of these convex bodies exhibit certain features, which signal the emergence of symmetry breaking and of an associated order parameter. I will illustrate this with a few paradigmatic examples of many body systems exhibiting symmetry breaking: the quantum Ising model, the classical Ising and Potts model in 2D at finite temperature and the ideal Bose gas in three dimensions at finite temperature. This quantum state based viewpoint on phase transitions provides a very intuitive and informative new way of drawing phase diagrams and constitutes a unique novel tool for studying exotic quantum phenomena.