Definition: Homogeneity and cone strict convexity

For a proper cone $K\subset V$, set

The cone is homogeneous if $\mathrm{Aut}(K)$ acts transitively on $\mathrm{int}K$. It is strictly convex in the cone sense if, for nonzero, non-proportional $u,v\in K$,

Equivalently, its boundary has no line segment joining points on different nonzero rays.