I am searching for the paper Bonsall, F. F. (1991). A General Atomic Decomposition Theorem and Banach’s Closed Range Theorem. The Quarterly Journal of Mathematics, 42 9–14. for the proof of $$ \inf\{t > 0: x \in t \text{conv}(A)\} = \inf\left\{ \sum_{a \in A} c_a: x = \sum_{a \in A} c_a a, \ c_a \ge 0 \ \forall a \in A\right\}, $$ where $A \subset \mathbb R^p$ is a compact subset whose elements are the extreme points of conv$(A)$, i.e. $a \not\in \text{conv}(A \setminus \{a\})$ for all $a \in A$ and the centroid of conv$(A)$ is the origin. (This is from The Convex Geometry of Linear Inverse Problems by Venkat Chandrasekaran, Benjamin Recht, Pablo A. Parrilo, and Alan S. Willsky)
I know that by Caratheodory theorem, one can represent the convex hull of $A$ as convex combinations of elements in $A$ with at most $p + 1$ summands, so I can write $$ \text{LHS} = \inf\left\{ r > 0: \frac{x}{r} = \sum_{k = 1}^{N} c_k a_k, \ N \le p + 1, \begin{array}{c} (c_k)_{k = 1}^{N} \subset [0, 1], \\ (a_k)_{k = 1}^{N} \subset A, \end{array} \sum_{k = 1}^{N} c_k = 1 \right\} $$ And I know that the LHS is $\infty$ (per convention $\inf(\emptyset) = \infty$) if $x$ is not in the affine hull of conv$(A)$. As conv$(A)$ is convex, aff$($conv$(A)) = $span$(A)$.
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