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. Properties Length and area Every bounded convex curve is a rectifiable curve, meaning that it has a well-defined finite arc length, and can be approximated in length by a sequence of inscribed polygonal chains. For closed convex curves, the length may be given by a form of the Crofton formula as times the average length of its projections onto lines. It is also possible to approximate the area of the convex hull of a convex curve by a sequence of inscribed convex polygons. For any integer , the most accurate approximating -gon has the property that each vertex has a supporting line parallel to the line through its two neighboring vertices. As Archimedes already knew, if two convex curves have the same endpoint, and one of the two curves lies between the other and the line through their endpoints, then the inner curve is shorter than the outer one. According to Newton's theorem about ovals, the area cut off from an infinitely differentiable convex curve by a line cannot be an algebraic function of the coefficients of the line. It is not possible for a short strictly convex curve to pass through many points of the integer lattice. If the curve has length , then according to a theorem of Vojtěch Jarník, the number of lattice points that it can