Let $X$ be a compact metric space and identify the set $P(X)$ of Borel probabilities measures with a convex subset $K_X$ of $C(X)'$ equipped with the weak$^{\star}$ topology.
Does $K_X$ satsify: $relcl(relint(K_X))=K_X$? (Where $relint(K_X)$ is its relative interior on $P(X)$ and $relcl$ is its relative closure)?
The affine subspace spanned by $K_X$ is $\{f \in C(X)’,\, f(1)=1\}$ (use the measure characterization of $C(X)’$).
Let $\mu$ be in the relative interior of $K_X$. This means that for all $p \in C(X)’$ with $p(X)=1$, there exists arbitrarily small $0 < \lambda < 1$ such that $(1+\lambda)\mu-\lambda p$ and $(1-\lambda)\mu + \lambda p$ are probability measures. (*)
In particular, for all $p \in C(X)’$, $p(1)=1$, $A \subset X$, if $\mu(A)=0$, then $p(A)=0$. Therefore, every point of $X$ is an atom for $\mu$, hence $X$ is countable.
So if $X$ is uncountable, the relative interior of $K_X$ is empty.
Assuming now $X$ to be countable, (*) yields that for all $p \in C(X)’$, if $p(X)=1$ then $dp=fd\mu$ for some bounded function $f: X \rightarrow \mathbb{R}$. So the same property holds if $p(X) \neq 0$ (scaling), thus for any $p$ (if $p(X)=0$ consider $p+\mu$).
This cannot hold if $X$ is infinite, regardless of topology.
So $K_X$ has nonempty relative interior iff $X$ finite iff it is regular.