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A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a simple polygon whose interior is a convex set.^{[1]} 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.
A polygon which is not convex is called concave.
The following properties of a simple polygon are all equivalent to convexity:
Additional properties of convex polygons include:
Every polygon inscribed in a circle (such that all vertices of the polygon touch the circle), if not self-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 nondegenerate triangle is strictly convex.
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