In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This definition is immediately generalizable to any real, or complex, vector space. Intuitively, if one thinks of S as a region surrounded by a wall, S is a star domain if one can find a vantage point s_0 in S from which any point s in S is within line-of-sight. A similar, but distinct, concept is that of a radial set.

Definition

Given two points x and y in a vector space X (such as Euclidean space \R^n), the convex hull of {x, y} is called the and it is denoted by where for every vector z. A subset S of a vector space X is said to be s_0 \in S if for every s \in S, the closed interval A set S is and is called a if there exists some point s_0 \in S such that S is star-shaped at s_0. A set that is star-shaped at the origin is sometimes called a. Such sets are closely related to Minkowski functionals.

Examples

Properties