nLab star domain




A star-shaped domain is a subset UU of an appropriate ambient space for which there exists a basepoint that can be connected to every other point by a “straight line in UU”. Such a subset is a contractible space, using the contraction that travels each point back to the basepoint along the given lines.


For XX a real affine space, a star domain about a point xXx\in X is an inhabited subset UXU \subset X such that with yXy \in X, the straight line segment connecting xx with yy in XX is also contained in UU.

For VV a vector space, a star domain about the origin is an inhabited subset UVU \subset V that is radial.

These definitions can be modified in various obvious ways. For example, a star shaped neighbourhood of a point xx in an affine space XX is an open neighbourhood UXU \subset X of xx that is a star domain about xx. 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 KK, where if vv is a vertex of KK, then the open star of vv is the union of the interiors in |K|{|K|} of all the simplices containing vv. 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.

Last revised on October 28, 2021 at 17:52:09. See the history of this page for a list of all contributions to it.