convex set in the Euclidean plane. Not every convex set has a connected boundary, but when it does, the whole boundary is an example of a convex curve. When a bounded convex set in the plane is not a line segment, its boundary forms a simple closed convex curve. By the Jordan curve theorem, a simple closed curve divides the plane into interior and exterior regions, and another equivalent definition of a closed convex curve is that it is a simple closed curve whose union with its interior is a convex set. Examples of open and unbounded convex curves include the graphs of convex functions. Again, these are boundaries of convex sets, the epigraphs of the same functions. This definition is equivalent to the definition of convex curves from support lines. Every convex curve, defined as a curve with a support line through each point, is a subset of the boundary of its own convex hull. Every connected subset of the boundary of a convex set has a support line through each of its points. Intersection with lines For a convex curve, every line in the plane intersects the curve in one of four ways: its intersection can be the empty set, a single point, a pair of points, or an interval. In the cases where a closed curve intersects in a single point or an interval, the line is a supporting line. This can be used as an alternative definition of the convex curves: they are the Jordan curves (connected simple curves) for which every intersection with a line has one of these four types. This definition can be used to generalize convex curves from the Euclidean plane to certain other linear spaces such as the real projective plane. In these spaces, like in the Euclidean plane, any curve with only these restricted line intersections has a supporting line for each point. Strict convexity The strictly convex curves again have many equivalent definitions. They are the convex curves that do not contain any line segments. They are the curves for which every intersection of the curve with a line consists of at most two points. They are the curves that can be formed as a connected subset of the boundary of a strictly convex set. Here, a set is strictly convex if every point of its boundary is an extreme point of the set, the unique maximizer of some