Decomposition of the Measure in the Integral Representation of Piecewise Convex Curves

Summary

The notions of a convex arc and piecewise convex curve in the plane generalize the notion of a convex curve, the latter is usually defined as the boundary of a planar compact convex set with nonempty interior. The integral representation of a piecewise convex curve through a Riemann-Stieltjes integral with a corresponding one-dimensional measure is studied. It is shown that the Minkowski operations known from the convex sets can be generalized to piecewise convex curves. It is shown that the decomposition of the measure in the integral representation of the piecewise convex curve leads to a decomposition of the piecewise convex curve into a sum of corresponding piecewise convex curves. On this base, applying the natural decomposition of the one-dimensional measure into an absolutely continuous function, a jump function, and a singular function, the structure of a piecewise convex curve is investigated. As some curious consequences, the existence of polygons with infinitely many sides and no vertices, and polygons with infinitely many vertices and no sides is shown.