CONVEX TRANSITIVE NORMS ON SPACES OF CONTINUOUS FUNCTIONS

Abstract

A norm on a Banach space $X$ is called maximal when no equivalent norm has a larger group of isometries. If, besides this, there is no equivalent norm with the same isometries (apart from its scalar multiples), the norm is said to be uniquely maximal, which is equivalent to the convex-transitivity of $X$: the convex hull of the orbits under the action of the isometry group on the unit sphere is dense in the unit ball of $X$. The main result of the paper is that the complex $C_0(\Omega)$ is convex-transitive in its natural supremum norm if $\Omega$ is a connected manifold (without boundary). As a complement, it is shown that if $\Omega$ is a connected manifold of dimension at least two, then the diameter norm is convex transitive on the corresponding space of real functions.