A star-shaped domain is a subset of an appropriate ambient space for which there exists a basepoint that can be connected to every other point by a “straight line in ”. Such a subset is a contractible space, using the contraction that travels each point back to the basepoint along the given lines.
These definitions can be modified in various obvious ways. For example, a star shaped neighbourhood of a point in an affine space is an open neighbourhood of that is a star domain about . Or, a subset is a star domain if it is a star domain about one of its points.
A useful special case pertains to a simplicial complex , where if is a vertex of , then the open star of is the union of the interiors in of all the simplices containing . Open stars of vertices provide a good open cover of a simplicial complex.
A convex set is the same as a set that is a star domain about each of its points.