Convex-transitivity and function spaces

It is shown that if X is a convex-transitive Banach space and 1⩽p<∞, then Lp([0,1],X) and are convex-transitive. Here is the closed linear span of the simple functions in the Bochner space L∞([0,1],X). If H is an infinite-dimensional Hilbert space and C0(L) is convex-transitive, then C0(L,H) is convex-transitive. Some new fairly concrete examples of convex-transitive spaces are provided.