Extremal Homogeneous Polynomials on Real Normed Spaces Original Research Article

IfPis a continuousm-homogeneous polynomial on a real normed space andPis the associated symmetricm-linear form, the ratio ‖P‖/‖P‖ always lies between 1 andmm/m!. We show that, as in the complex case investigated by Sarantopoulos (1987,Proc. Amer. Math. Soc.99, 340–346), there arePʹs for which ‖P‖/‖P‖=mm/m! and for whichPachieves norm if and only if the normed space contains an isometric copy of ℓm1. However, unlike the complex case, we find a plentiful supply of such polynomials providedm⩾4.