Convex curve

Bounded convex curves have a well-defined length, which can be obtained by approximating them with polygons, or from the average length of their projections onto a line. The maximum number of grid points that can belong to a single curve is controlled by its length. The points at which a convex curve has a unique supporting line are dense within the curve, and the distance of these lines from the origin defines a continuous support function. A smooth simple closed curve is convex if and only if its curvature has a consistent sign, which happens if and only if its total curvature equals its total absolute curvature.

Definitions

Archimedes, in his On the Sphere and Cylinder, defines convex arcs as the plane curves that lie on one side of the line through their two endpoints, and for which all chords touch the same side of the curve. This may have been the first formal definition of any notion of convexity, although convex polygons and convex polyhedra were already long known before Archimedes. For the next two millennia, there was little study of convexity: its in-depth investigation began again only in the 19th century, when Augustin-Louis Cauchy and others began using mathematical analysis instead of algebraic methods to put calculus on a more rigorous footing.

Many other equivalent definitions for the convex curves are possible, as detailed below. Convex curves have also been defined by their supporting lines, by the sets they form boundaries of, and by their intersections with lines. In order to distinguish closed convex curves from curves that are not closed, the closed convex curves have sometimes also been called convex loops, and convex curves that are not closed have also been called convex arcs.

Background concepts

A plane curve is the image of any continuous function from an interval to the Euclidean plane. Intuitively, it is a set of points that could be traced out by a moving point. More specifically, smooth curves generally at least require that the function from the interval to the plane be continuously differentiable, and in some contexts are defined to require higher derivatives. The function parameterizing a smooth curve is often assumed to be regular, meaning that its derivative stays away from zero; intuitively, the moving point never slows to a halt or reverses direction. Each interior point of a smooth curve has a tangent line. If, in addition, the second derivative exists everywhere, then each of these points has a well-defined curvature.

A plane curve is closed if the two endpoints of the interval are mapped to the same point in the plane, and it is simple if no other two points coincide. Less commonly, a simple plane curve may be said to be open if it is topologically equivalent to a line, neither having an endpoint nor forming any limiting point that does not belong to it, and dividing the plane into two unbounded regions. However, this terminology is ambiguous as other sources refer to a curve with two distinct endpoints as an open curve. Here, we use the topological-line meaning of an open curve.

Supporting lines

A supporting line is a line containing at least one point of the curve, for which the curve is contained in one of the two half-planes bounded by the line. A plane curve is called convex if it has a supporting line through each of its points. For example, the graph of a convex function has a supporting line below the graph through each of its points. More strongly, at the points where the function has a derivative, there is exactly one supporting line, the tangent line.

Supporting lines and tangent lines are not the same thing, but for convex curves, every tangent line is a supporting line. At a point of a curve where a tangent line exists, there can only be one supporting line, the tangent line. Therefore, a smooth curve is convex if it lies on one side of each of its tangent lines. This may be used as an equivalent definition of convexity for smooth curves, or more generally for piecewise smooth curves.

Boundaries of convex sets

A convex curve may be alternatively defined as a connected subset of the boundary of a 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 linear function. As the boundaries of strictly convex sets, these are the curves that lie in convex position, meaning that none of their points can be a convex combination of any other subset of its points.

Closed strictly convex curves can be defined as the simple closed curves that are locally equivalent (under an appropriate coordinate transformation) to the graphs of strictly convex functions. This means that, at each point of the curve, there is a neighborhood of the points and a system of Cartesian coordinates within that neighborhood such that, within that neighborhood, the curve coincides with the graph of a strictly convex function.

Symmetry

Smooth closed convex curves with an axis of symmetry, such as an ellipse or Moss's egg, may sometimes be called ovals. However, the same word has also been used to describe the sets for which each point has a unique line disjoint from the rest of the set, especially in the context of ovals in finite projective geometry. In Euclidean geometry these are the smooth strictly convex closed curves, without any requirement of symmetry.