Extremal homogeneous polynomials on real normed spaces

If P is a continuous nz-homogeneous polynomial on a real normed space and P is the associated symmetric m-linear form, the ratio \\P\\/\\P\\ always li es between 1 and m(m)/m!. We show that, as in the complex case investigated by Sarantopoulos (1987, Proc. Amer. Math. Sec. 99, 340-346), there are P's for which \\P\\/\\P\\= m(m)/m! and for which ??P achieves norm if and only if the normed space contains an isometric copy of l(1)(m). However, unlike the complex case, we find a plentiful supply of such polynomials provided m greater than or equal to 4. (C) 1999 Academic Press.