Spans and convex combinations of boundary-valued continuous functions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909839507062784 |
|---|---|
| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | 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 $E$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07857 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spans and convex combinations of boundary-valued continuous functions Chirvasitu, Alexandru Functional Analysis General Topology 46E15, 52A21, 55M25, 54F45, 54C65, 52A07, 54D15, 55P05 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 $E$. |
| title | Spans and convex combinations of boundary-valued continuous functions |
| topic | Functional Analysis General Topology 46E15, 52A21, 55M25, 54F45, 54C65, 52A07, 54D15, 55P05 |
| url | https://arxiv.org/abs/2510.07857 |