= Strictly positive barycentre cone lemma
{title2=$\{\mathbb E[\nu X]:\nu>0,\ \mathbb E[\nu\|X\|]<\infty\}=\operatorname{ri}K$}
For a finite-dimensional <random vector> $X$, let $K$ be the closed cone generated by the <support of a probability distribution> of $X$. Every integrable strictly positively weighted barycentre lies in the <relative interior> of $K$: every nonzero supporting functional on its span is positive on a set of positive probability. Conversely, put $w=(1+\|X\|)^{-1}$ and $D=\{\mathbb E[w fX]:0\leq f\in L^\infty\}$. Its dual cone equals that of $K$, so $\overline D=K$ and $\operatorname{ri}K\subseteq D$. For $z\in\operatorname{ri}K$, subtract a sufficiently small positive multiple of $\mathbb E[wX]$ and represent the remainder using $D$; adding the constant positive weight back gives $\nu=w(f+\varepsilon)>0$.
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