Conic hull ($\mathrm{cone}(S)$)

The set of all finite conic combinations of elements of $S$, including zero. It is the smallest convex cone containing $S$. It need not be closed even when $S$ is closed: the closed set $S=\{(t,1):t\ge 1\}$ generates vectors converging to $(1,0)$ via $(t,1)/t$, but no nonzero generated vector has second coordinate zero.

A finitely generated cone is a conic hull of a finite set of vectors. Every point admits a representation using linearly independent active generators, by eliminating dependence while keeping coefficients nonnegative.

A finitely generated cone is a closed convex cone. For a convergent sequence of represented vectors, use linearly independent active generators and pass to a subsequence with the same active set. Its coefficients converge through a fixed left inverse and retain nonnegativity. General linear images of closed cones need not be closed; finiteness of the generators supplies the missing argument.

A finite sum ${\sum }_j{\lambda }_j{s}_j$ with ${\lambda }_j\ge 0$. Unlike a convex combination, the coefficients need not sum to one. Zero coefficients and the empty sum are allowed.