Definition:Convex Set (Vector Space)/Line Segment

It has been suggested that this page be renamed. In particular: See talk To discuss this page in more detail, feel free to use the talk page.

Definition

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

Let $x, y \in V$.

The set:

$\set {t x + \paren {1 - t} y: t \in \closedint 0 1}$

is called the (straight) line segment joining $x$ and $y$.

A convex set can thus be described as a set containing all straight line segments between its elements.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): convex set
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convex set