Convex polygon

Properties

• 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.

• 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 the intersection of every three of them is nonempty, then the whole collection has a nonempty intersection.
• 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. If the polygons are closed and at least one of them is compact, then there are even two parallel separator lines.
• Inscribed triangle property: Of all triangles contained in a convex polygon, there exists a triangle with a maximal area whose vertices are all polygon vertices.
• Inscribing triangle property: every convex polygon with area A can be inscribed in a triangle of area at most equal to 2A. Equality holds for a parallelogram.
• Inscribed/inscribing rectangles property: For every convex body C in the plane, we can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed about C and the positive homothety ratio is at most 2 and.
• The mean width of a convex polygon is equal to its perimeter divided by pi. So its width is the diameter of a circle with the same perimeter as the polygon.

Strict convexity

• Every internal angle is strictly less than 180 degrees.
• Every line segment between two points in the interior, or between two points on the boundary but not on the same edge, is strictly interior to the polygon.
• For each edge, the interior points and the boundary points not contained in the edge are on the same side of the line that the edge defines.
• The angle at each vertex contains all other vertices in its interior.