Convex set
![Illustration of a convex set which looks somewhat like a deformed circle. The (black) line segment joining points x and y lies completely within the (green) set. Since this is true for any points x and y within the set that we might choose, the set is convex.](/Images/godic/202501/06/Convex_polygon_illustration12305.png")
![Illustration of a non-convex set. Since the red part of the (black and red) line-segment joining the points x and y lies outside of the (green) set, the set is non-convex.](/Images/godic/202501/06/Convex_polygon_illustration22305.png")
![A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set.](/Images/godic/202501/06/Convex_supergraph2305.png")
![Minkowski addition of sets. The sum of the squares Q1=[0,1]2 and Q2=[1,2]2 is the square Q1+Q2=[1,3]2.](/Images/godic/202501/06/Minkowski_sum2305.png")
In Euclidean space, a convex set is the region such that, for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the region. For example, a solid cube is a convex set, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex. A convex curve forms the boundary of a convex set.