Is a conic hull of any arbitrary set not a strictly convex set?

Question

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?

in the figure, the conic hull is the union of origin and an open set. So, it is strictly convex (no line on the boundary)?

Answer 1

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.