Spans and convex combinations of boundary-valued continuous functions

For an (n≥2)-dimensional real Banach space E with unit ball E_≤1 and a topological space X arbitrary elements in C(X,E_≤1) are always expressible as linear combinations of at most three functions valued in the unit sphere ∂E_≤1. On the other hand, for normal X , C(X,E_≤1) can only be the convex hull of C(X,∂E_≤1) if the covering dimension of X is strictly smaller than dim⁡E. A variant of this remark is the characterization of normal X with dim⁡X<dim⁡E as precisely those for which C(X,E_≤1) is the convex hull of nowhere-vanishing continuous X→E_≤1 or, equivalently, that of continuous functions X→E_[r,1], r∈(0,1) valued in arbitrarily thin spherical shells. This extends a number of results due to Peck, Cantwell, Bogachev, Mena-Jurado, Navarro-Pascual and Jiménez-Vargas and others revolving around the realizability of the unit ball of C(X,E) as a convex hull of its extreme points for strictly convex and/or complex E .