star domain

For VV a vector space, a star domain about the origin is an inhabited subset UVU \subset V such that with vUv \in U and s[0,1]s \in [0,1] also svUs v \in U.

More generally, 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.

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.

Revised on September 21, 2015 12:20:02 by Todd Trimble (