# A Caratheodory-like result for infinite-dimensional simplices

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 $$K$$.