Name of a shape that is intersected once by each ray that starts at a given point

Question

Is there a particular name for a shape that is intersected exactly once by each ray that starts at a given point?

To illustrate: I'm looking for a name for shapes like the left one in this image:

(This is for 2D, but the same could be applied to surfaces in higher dimensions)

I thought about a word describing the shape and its relation to the point, for example, that the shape is "convex referring to a single point", but this does not seem to be appropriate. One could possibly just call it a "circle (or sphere) with varying radius"; does anyone know if there is a more adequate name for this?

Answer 1

The set that you describe is the boundary of a star-shaped domain (also called star domain, star-convex set and radially convex set). The formal definition is as follows:

A set $S$ in the Euclidean space is said to be star-shaped if there exists some point $x_0\in S$ such that for any $x\in S$ the line segment $x_0x$ is contained in $S.$

See the following link for more information:

http://en.wikipedia.org/wiki/Star_domain

Since you consider only the curve (or surface) limiting such a set, the usual is the boundary of the set.

See

http://en.wikipedia.org/wiki/Boundary_%28topology%29

for more information about the boundary of a set.