Geometry of Three-Homogeneous Polynomials on Real Hilbert Spaces

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Let H be a two-dimensional real Hilbert space. We give a characterisation of the extreme and the smooth points of the unit ball of the space of three-homogeneous polynomials on H in terms of the coefficients of the polynomial. We also determine the smooth points of the unit ball of the predual of the space of three-homogeneous polynomials.