nLab radial set

In the theory of locally convex topological vector spaces, certain types of subset turn up more frequently than others. One such is radial subsets. In a radial subset, and line segment from the origin to a point in the subset is contained in the subset. Thus the condition is similar to that of being convex, except that one of the end-points in the convexity condition is constrained to be the origin. Thus a convex set containing the origin is automatically radial.

The precise definition is as follows.

Definition

Let $E$ be a locally convex topological vector space. A subset $A$ of $E$ is radial if $r A \subseteq A$ for each $r \in [0,1]$.

A radial set is clearly a star-shaped domain.