= Solution
The <convex hull> of $S\subseteq\mathbb R^d$ is
$$
\operatorname{conv}S
=\left\{\sum_{j=1}^m\lambda_jv_j:
m<\infty,\ v_j\in S,\ \lambda_j\geq0,\
\sum_{j=1}^m\lambda_j=1\right\}.
$$
Thus $v$ is a convex combination of $v_1,\ldots,v_m$ when it has precisely this form. Equivalently, $\operatorname{conv}S$ is the smallest <convex set> containing $S$. The <Caratheodory theorem> sharpens the finite-combination description in $\mathbb R^d$: every point of $\operatorname{conv}S$ is a convex combination of at most $d+1$ points of $S$.
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