 Strictly positive barycentre cone lemma

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 $w=(1+\Vert X\Vert {)}^{-1}$ and $D=\{\mathbb{E}[wfX]:0\le f\in L^{\infty }\}$. Its dual cone equals that of $K$, so $\overline{D}=K$ and $\mathrm{ri}K\subseteq D$. For $z\in \mathrm{ri}K$, subtract a sufficiently small positive multiple of $\mathbb{E}[wX]$ and represent the remainder using $D$; adding the constant positive weight back gives $\nu =w(f+\epsilon )>0$.