Geometry of 2-homogeneous polynomials on p spaces, 1

We characterise extreme 2-homogeneous polynomials of diagonal form on p with p > 2 and give a class of examples of smooth points. We describe the unit ball of P(2 2 p) for both the cases p >2 and p <2. We apply the results thus found to study the geometry of the predual of P(2 2 p) and to show that for p >2 an extreme 2-homogeneous polynomial on 2 p remains extreme when regarded as a polynomial on p.  2002 Elsevier Science (USA). All rights reserved.