a norm on $X$ that induces the topology that $X$ inherits from $V$. Assume additionally that $X$ contains the intersection of the tangent cone of $C$ at $p$ with an open subset of $V$ that contains $v$. Then there is a curve $c\colon[0,+\infty)\to C$ such that $$c(0)=p\quad \textrm{and}\quad dc(t)/dt|_{t=0+}=v$$ if, and only if, $\theta(v){\geqslant}0$ for all $\theta$ in $V^*$ for which $\theta(p)=\inf_{q\in C}\theta(q)$, that is, for all continuous linear functionals $\theta$ that attain their minimum within $C$ at $p$.

We construct a sequence $v_1,v_2,\dots$ in the tangent cone to $C$ at $p$ converging to $v$, $v_i\to v$. For $i\in {\mathbb{N}}$, we take the balls $B_i$ of radius $1/i$ centered at $v$ with respect to the norm in $X$. These must all intersect that tangent cone because $v$ is in its topological closure. The tangent cone is contained in $X$, so we take $v_i$ to be any point in its intersection with $B_i$. From this point on, the rest of the proof of Theorem \[thm:characterization\] works.

Families of positive measures and probabilities {#sec:families}
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Characterization {#sec:characterization}
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The following theorem is the main result of this section and follows from Corollary \[cor:useful\] in a way that will be explained below. Important 