Let H be a two-dimensional complex Hilbert space and P( 3 H ) the space of 3-homogeneous polynomials on H . We give a characterization of the extreme points of its unit ball, B P( 3 H ) , from which we deduce that the unit sphere of P( 3 H ) is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of B P( 3 H ) remains extreme as considered as an element of B L( 3 H ) . Finally we make a few remarks about the geometry of the unit ball of the predual of P( 3 H ) and give a characterization of its smooth points.
Notation, terminology and preliminary resultsGiven n (real or complex) Banach spaces X 1 , . . . , X n we denote by X 1 ⊗· · ·⊗ X n their tensor product and by π the projective norm. If X 1⊗π · · ·⊗ π X n is the completion of X 1 ⊗· · · ⊗ X n under the projective norm, then we have (X 1⊗π · · ·⊗ π X n ) * = L( n X 1 , . . . , X n ), the space of continuous n-linear forms on X 1 × · · · × X n endowed with the supremum norm. However, as remarked in [6], most of the times the multilinear theory is far from being just a simple