An example of a convex polygon: aregular pentagon.
Ingeometry, aconvex polygon is apolygon that is theboundary of aconvex set. This means that theline 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 asimple polygon (notself-intersecting).[1] Equivalently, a polygon is convex if everyline that does not contain any edge intersects the polygon in at most two points.
A convex polygon isstrictly convex if no line contains more than two vertices of the polygon. In a convex polygon, allinterior angles are less thanor equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees.
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 theconvex 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 (with a gap between them).
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.[2]
Inscribing triangle property: every convex polygon with area can be inscribed in a triangle of area at most equal to. Equality holds (exclusively) for aparallelogram.[3]
Inscribed/inscribing rectangles property: For every convex body in the plane, we can inscribe a rectangle in such that ahomothetic copy of is circumscribed about and the positive homothety ratio is at most 2 and.[4]
Themean width of a convex polygon is equal to its perimeter divided by, thus the diameter of a circle with the same perimeter as the polygon.[5]
Every polygon inscribed in a circle (such that all vertices of the polygon touch the circle), if notself-intersecting, is convex. However, not every convex polygon can be inscribed in a circle.
The following properties of a simple polygon are all equivalent to 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 (except at its endpoints if they are on the edges).
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 (except the given vertex and the two adjacent vertices).
^Chandran, Sharat; Mount, David M. (1992). "A parallel algorithm for enclosed and enclosing triangles".International Journal of Computational Geometry & Applications.2 (2):191–214.doi:10.1142/S0218195992000123.MR1168956.
