Radially open convex set

A subset $C$ of a real vector space is radially open when, for each $x\in C$ and each vector $v$, the points $x+tv$ remain in $C$ for all sufficiently small real $t$. No ambient topology is required. If $C$ is also convex and contains zero, it is absorbing and its Minkowski functional $p_C(x)=\inf \{s>0:x\in sC\}$ is finite and sublinear, with $C=\{p_C<1\}$. Convexity gives subadditivity; radial openness permits a slight dilation of any point of $C$, giving the strict inequality.