Is a conic hull of any arbitrary set not a strictly convex set? A conic hull of an arbitrary set can be a strictly convex set? Please look at the figure below, the set is a triangle which is not closed. What is its conic hull? ?

EDITED:

The conic hull of an open set is the union of an open set and the origin, and this is strictly convex. But a convex cone that contains one of its boundary points other that $0$ is not strictly convex, so these are essentially the only counterexamples. The conic hull $\text{coni}(S)$ of a set $S$ is strictly convex if and only if $\text{coni}(S)$ is $\\{0\\}$ or the conic hull of an open set.