However, not every convex polygon can be inscribed in a circle. Įvery polygon inscribed in a circle (such that all vertices of the polygon touch the circle), if not self-intersecting, is convex. So its width is the diameter of a circle with the same perimeter as the polygon.
If the polygons are closed and at least one of them is compact, then there are even two parallel separator lines (with a gap between them). Hyperplane separation theorem: Any two convex polygons with no points in common have a separator line.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. Krein–Milman theorem: A convex polygon is the convex hull of its 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.A convex polygon may be triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices.The intersection of two convex polygons is a convex polygon.The polygon is the convex hull of its edges.Īdditional properties of convex polygons include:.The angle at each vertex contains all other vertices in its edges and interior.For each edge, the interior points are all on the same side of the line that the edge defines.The polygon is entirely contained in a closed half-plane defined by each of its edges.Every point on every line segment between two points inside or on the boundary of the polygon remains inside or on the boundary.