by the seminorm $|\cdot|'_{\overline U,n}$, which is nondegenerate within $X$. We also observe that $V$ is locally convex. The set $X$ contains all signed Radon measures $\nu$ with $|\nu|_{\overline U,n}'<2|\eta|'_{\overline U,n}$, so in particular it contains the intersection of the tangent cone of $C$ at $p$ with the set of distributions $\xi$ with $|\xi|_{\overline U,n}'<2|\eta|'_{\overline U,n}$. This set of distributions is an open set that contains $v$. So we can apply Corollary \[cor:useful\], which is then equivalent to the following assertion: A family $(\mu_t)_{t{\geqslant}0}$ as in the statement of Theorem \[thm:families\] exists if, and only if, $\langle\eta,f\rangle{\geqslant}0$ for all functions $f\in C^\infty_c(P)$ such that $$\label{eq:minimum}
  \int f\,d\mu{\leqslant}\int f\,d\nu\quad \textrm{for all}\quad \nu\in\mathscr M_0.$$

To finish the proof of the first part of Theorem \[thm:families\], we just need to understand these functions $f$. Let $f\in C^\infty_c(P)$ satisfy . Clearly, the support of $\mu$ must be contained in the set of points $p$ such that $f(p)=\inf_{q\in P}f(q)$, so in order to prove the theorem we just need to argue that this infimum equals zero, or equivalently, that $\int f\,d\mu=0$. Taking $\nu_a=a\mu\in\mathscr M_0$ for $a>0$, it is easy to see that if $\inf_{q\in P}f(q)\neq 0$, then $a$ can be taken appropriately so that $\int 