in a context in which such “movement” can be extremely singular.

An initial version of the results presented in Section \[sec:families\] was posted previously in the arXiv in [@myderivatives; @myvariationalstructure]. The theory was since completely redeveloped from a different point of view until it attained the form presented here, which is also considerably simpler and more general.

#### Acknowledgements.

The author is very grateful to Patrick Bernard, Jaime Bustillo, and Stefan Suhr for their encouragement, support, suggestions, and numerous discussions.

Curves in convex sets {#sec:main}
=====================

Consider a locally-convex topological vector space $V$ with topological dual space $V^*$. Let $C$ be a closed and convex subset of $V$.

We say that a subset $A$ of $V$ is a *cone* if $ra$ is contained in $A$ for every $r>0$ and for every $a$ in $A$. The set $A$ is a *closed cone* if it is a cone and is closed in the topology of $V$.

Let $p$ be a point in the closed, convex set $C$. The *tangent cone* of $C$ at $p$ is the set of vectors $v$ such that $p+tv$ is contained in $C$ for some $t>0$. In other words, the tangent cone is the set $\mathbb R_{>0}(C-p)$. This 