Convex Area on a Plane

a part of a plane such that a section connecting any two of its points is entirely contained within that part of the plane. Any connecting part of the boundary of a convex area is called a convex curve. Examples of such curves are circles, ellipses, parabolas, triangles, any arc of a circle, straight lines, and sections of straight lines. At least one straight line of support that shares a common point (or section) with the boundary of the area but does not intersect the boundary passes through each point of the boundary of a convex area on a plane.

Convex areas on a plane may be of four types: finite (the boundary is a closed convex curve), infinite (the boundary is one infinite curve—for example, a convex area bounded by a parabola), an infinite strip (the boundary is a pair of parallel straight lines), or the entire plane. A convex area can be given by means of a support function, which expresses the distance from the beginning of the coordinates to the straight line of support as a function from the external standard to the convex area (that is, the unit vector perpendicular to the straight line of support and directed toward the one of the two semiplanes defined by this straight line in which there are no points of the convex area). A convex area on a plane is a partial (two-dimensional) case of n-dimensional convex areas, which are studied in the geometry of convex bodies.