Let $K$ be a compact metric space; $\Delta K$ be the set of Borel probability measures on $K$ endowed with the weak* topology; $X$ be a closed subset of $\Delta K$; and $x_0 \in \overline{\text{co}} X$.
Question: Does there necessarily exist some closed $Y\subseteq X$ such that $x_0 \in \overline{\text{co}} Y$ and $|Y|\leq |K|$?
As a starting observation, this is certainly true if $K$ is not a countably infinite space. Indeed:
- Kuratowski’s theorem tells us that both $K$ and $X$ must be either finite, of continuum cardinality, or countably infinite.
- If $K$ is finite, the result follows directly from Carathéodory’s theorem.
- If $K$ uncountable, and so of continuum cardinality, then $|X| \leq |K|$. So $Y=X$ will do.
So, we can specialize the question to the case that $K$ is a countably infinite, compact metric space.
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A related question whose answer I don't know for the same setup:
2) Does there necessarily exist some Borel $Y\subseteq X$ and Borel probability measure $p$ on $Y$ such that $x_0$ is the barycentre of $p$ and $|Y|\leq |K|$?
For any given $K$, a positive answer to the first question implies a positive answer to the second question (using the same witnessing $Y$). Therefore, for the second question too we may restrict attention to countably infinite $K$.
