the same set of points results, but its curvature is A smooth simple closed curve, with a regular parameterization, is convex if and only if its curvature has a consistent sign: always non-negative, or always Every smooth simple closed curve with strictly positive (or strictly negative) curvature is strictly convex, but some strictly convex curves can have points with curvature The total absolute curvature of a smooth convex curve, is at It is exactly for closed convex curves, equalling the total curvature of these curves, and of any simple closed curve. For convex curves, the equality of total absolute curvature and total curvature follows from the fact that the curvature has a consistent sign. For closed curves that are not convex, the total absolute curvature is always greater and its excess can be used as a measure of how far from convex the curve is. More generally, by Fenchel's theorem, the total absolute curvature of a closed smooth space curve is at with equality only for convex plane By the Alexandrov theorem, a non-smooth convex curve has a second derivative, and therefore a well-defined curvature, almost everywhere. This means that the subset of points without a second derivative has measure zero in the curve. However, in other senses, the set of points with a second derivative can be small. In particular, for the graphs of generic non-smooth convex functions, it is a meager set, that is, a countable union of nowhere dense sets. Inscribed polygons The boundary of any convex polygon forms a convex curve (one that is a piecewise linear curve and not strictly convex). A polygon that is inscribed in any strictly convex curve, with its vertices in order along the curve, must be a convex polygon. The inscribed square problem is the problem of proving that every simple closed curve in the plane contains the four corners of a square. Although still unsolved in general, its solved cases include the convex curves. In connection with this problem, related problems of finding inscribed quadrilaterals have been studied for convex curves. A scaled and rotated copy of any rectangle or trapezoid can be inscribed in any given closed convex curve. When the curve is smooth, a scaled and rotated copy of any cyclic quadrilateral can be inscribed in it. However, the assumption of smoothness is necessary for this result, because some right kites cannot be inscribed in some obtuse