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Concept
Convex polygon
Summary
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points.
A strictly convex polygon is a convex polygon such that no line contains two of its edges. In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees.
The following properties of a simple polygon are all equivalent to convexity:
Every internal angle is strictly less than 180 degrees.
Every point on every line segment between two points inside or on the boundary of the polygon remains inside or on the boundary.
The polygon is entirely contained in a closed half-plane defined by each of its edges.
For each edge, the interior points are all on the same side of the line that the edge defines.
The angle at each vertex contains all other vertices in its edges and interior.
The polygon is the convex hull of its edges.
Additional properties of convex polygons include:
The intersection of two convex polygons is a convex polygon.
A convex polygon may be triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices.
Helly's theorem: For every collection of at least three convex polygons: if all intersections of all but one polygon are nonempty, then the intersection of all the polygons is nonempty.
Krein–Milman theorem: A convex polygon is the convex hull of its vertices. Thus it is fully defined by the set of its vertices, and one only needs the corners of the polygon to recover the entire polygon shape.
Hyperplane separation theorem: Any two convex polygons with no points in common have a separator line.
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