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Concept
Convex curve
Summary
In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions. Important subclasses of convex curves include the closed convex curves (the boundaries of bounded convex sets), the smooth curves that are convex, and the strictly convex curves, which have the additional property that each supporting line passes through a unique point of the 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.
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.
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