Convex curve

In geometry, a convex curve is a curve in the Euclidean plane which lies completely on one side of each and every one of its tangent lines.

The boundary of a convex set is always a convex curve.

Any straight line L divides the Euclidean plane to two half-planes whose union is the entire plane and whose intersection is L . We say that a curve C "lies on one side of L" if it is entirely contained in one of the half-planes. A plane curve is called convex if it lies on one side of each of its tangent lines. In other words, a convex curve is a curve that has a supporting line through each of its points.

A convex curve may be defined as the boundary of a convex set in the Euclidean plane. This definition is more restrictive than the definition in terms of tangent lines; in particular, with this definition, a convex curve can have no endpoints.

Sometimes, a looser definition is used, in which a convex curve is a curve that forms a subset of the boundary of a convex set. For this variation, a convex curve may have endpoints.

A strictly convex curve is a convex curve that does not contain any line segments. Equivalently, a strictly convex curve is a curve that intersects any line in at most two points, or a simple curve in convex position, meaning that none of its points is a convex combination of any other subset of its points.

Every convex curve that is the boundary of a closed convex set has a well-defined finite length. That is, these curves are a subset of the rectifiable curves.

According to the four-vertex theorem, every smooth convex curve that is the boundary of a closed convex set has at least four vertices, points that are local minima or local maxima of curvature.

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