In the case finite-dimensional, for example, $A\subset \mathbb{R^n}$ convex, we know that $\mathrm{cone}(A)$ the cone enerated by $A$ is defined by $$\mathrm{cone}(A):=\left\{ \left.\sum_{i=1}^{m}\alpha_{i}x_{i} \:\right|m\leq n+1,\: x_{i}\in A, \: \alpha_{i}\geq 0 \:\:\forall i=1,\ldots,m,\:\: \sum_{i=1}^{m}\alpha_{i}=1\right\}.$$
The Question: Let $X$ be a infinite-dimensional vector space and $A\subseteq X$ convex set. Is there a definition of $\mathrm{cone}(A)$ the cone generated by $A$?
You wrote down a characterization of the convex hull. You don't want $\sum_{i}\alpha_i=1$. If you delete this, and replace "$m\leq n+1$" by "$m$ is a positive integer", then you are good.