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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. The notion of a convex set can be generalized to other spaces as described below. == In vector spaces == Let be a vector space over the real numbers, or, more generally, some ordered field. This includes Euclidean spaces. A set in is said to be convex if, for all and in and all in the interval , the point also belongs to . In other words, every point on the line segment connecting and is in . This implies that a convex set in a real or complex topological vector space is path-connected, thus connected. Furthermore, is strictly convex if every point on the line segment connecting and other than the endpoints is inside the interior of . A set is called absolutely convex if it is convex and balanced. The convex subsets of (the set of real numbers) are simply the intervals of . Some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids. The Kepler-Poinsot polyhedra are examples of non-convex sets. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「convex set」の詳細全文を読む スポンサード リンク
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