# Definition:Star Convex Set

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## Definition

Let $V$ be a vector space over $\R$ or $\C$.

A subset $A \subseteq V$ is said to be **star convex** iff there exists $a \in A$ such that:

- $\forall x \in A: \forall t \in \left[{0 \,.\,.\, 1}\right]: t x + \left({1 - t}\right) a \in A$

### Star Center

The point $a \in A$ is called a **star center of $A$**.

A **star convex set** can thus be described as a set containing all line segments between the **star center** and an element of the set.

## Also known as

A **star convex set** is also called a **star domain**, a **star-like set**, a **star-shaped set**, or a **radially convex set**.

The hyphenated form **star-convex set** is also used.

## Also see

- Star Shaped Set, which is a different definition.

## Sources

- 1975: James R. Munkres:
*Topology*: $\S 53$ - 2001: Christian Berg:
*Kompleks funktionsteori*: $\S 3.1$