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Convex set

A set where the straight segment joining any two of its points never leaves the set.

A set S is convex if the straight segment joining any two points of S stays entirely inside S. Every disk is convex; a punctured disk is not, since the segment through the two points on opposite sides of the missing center passes through it. Convexity names the simplest way two points of a set can be joined without leaving it -- one straight segment -- and it is stronger than connectedness requires.

Examples

  • The disk |z|<3 is convex -- the straight segment between any two of its points stays inside it.

  • A punctured disk 0<|z-z_0|<r is not convex -- the segment through two points on opposite sides of z_0 passes through the missing center.

See also

Connected set Polygonally connected set Domain (of the plane)
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