Extremal multilinear forms on Banach spaces

Abstract:

Suppose that $L$ is a continuous symmetric $m$-linear form defined on a complex Banach space $E$, and $\hat L$ is the associated homogeneous polynomial. If \[ || L || = ({m^m}/m!)|| {\hat L} ||,\] we prove that $E$ contains an almost isometric copy of $l_m^1$. In particular if $E$ is an $m$-dimensional space, then $E$ is isometrically isomorphic to $l_m^1$. We also prove that the only examples of such extremal $L$ which achieve their norm are suitable "extensions" of a known example given by Nachbin.