Spans and convex combinations of boundary-valued continuous functions
Alexandru Chirvasitu
TL;DR
Problem: determine when the unit ball $C_b(X,E)_{\le 1}=C(X,E_{\le 1})$ can be reconstructed as the convex hull of boundary-valued maps $C_b(X,\partial E_{\le 1})$ and how this interacts with the dimensions of $X$ and the finite-dimensional $E$.\nApproach: introduce Krein–Milman invariants $\textsc{KM}^{\bullet}_{\Box;E}(X)$, derive dimension-based obstructions (e.g., $\textsc{KM}^{\mathrm{cvx}}_{\partial;E}(X)<\infty$ implies $\dim X<\dim E$) and establish a universal three-term linear decomposition: every $f\in C_b(X,E_{\le 1})$ is a linear combination of at most three $\partial E_{\le 1}$-valued maps, via a constructive scheme using bisected chords and degrees of sphere maps.\nContributions: generalize prior results by removing strict convexity/complex-structure assumptions, prove $\textsc{KM}^{\mathrm{spn}}_{\partial;E}(X)=3$ for $\dim E=n\ge2$, relate convex hulls to shells $E_{[r,1]}$, and characterize when $C_b(X,E_{\le 1})$ is the convex hull of nowhere-vanishing maps.\nSignificance: links convex-geometric realizability questions in function spaces with topological-dimension theory, extending the Peck–Cantwell–Bogachev–Mena-Jurado–Navarro-Pascual–Jim\'enez-Vargas lineage and providing sharp decomposition bounds for unit balls of $C(X,E)$.
Abstract
For an $(n\ge 2)$-dimensional real Banach space $E$ with unit ball $E_{\le 1}$ and a topological space $X$ arbitrary elements in $C(X,E_{\le 1})$ are always expressible as linear combinations of at most three functions valued in the unit sphere $\partial E_{\le 1}$. On the other hand, for normal $X$, $C(X,E_{\le 1})$ can only be the convex hull of $C(X,\partial E_{\le 1})$ 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 $C(X,E_{\le 1})$ is the convex hull of nowhere-vanishing continuous $X\to E_{\le 1}$ or, equivalently, that of continuous functions $X\to E_{[r,1]}$, $r\in (0,1)$ 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 $E$.