Since this implies the center of gravity is within the shape itself, this is a particular type of star domain. Some theorems about these are listed on Wikipedia. I stumbled across some information beyond Wikipedia since first answering.
Here it seems to suggest a slight generalization to not-necessarily-closed sets is preferred for applications to the geometry of numbers.
I found this idea is also used to define intersection bodies, which are the main tool for solving the Busemann-Petty problem. In this case, they are called star bodies, but it's assumed they are compact (hence closed), that their center is the origin, and that the maximum distance from the center is a continuous function of the angle of the ray out from the center (or rather, a continuous function of the point on the hypersphere parametrizing half-lines through the center). The work on this problem introduces many interesting results on star domains.