Piecewise Convex Curves and their Integral Representation

Abstract

A convex arc in the plane is introduced as an oriented arc G satisfying the following condition: For any three of its points c1 < c2 < c3 the triangle c1c2c3 is counter-clockwise oriented. It is proved that each such arc G is a closed and connected subset of the boundary of the set FG being the convex hull of G. It is shown that the convex arcs are rectifyable and admit a representation in the natural parameter by the Riemann-Stieltjes integral with respect to an increasing, nonnegative and continuous from the right function s+. Further it is shown that the obtained representation relates to the support function of the set FG. Concerning the reverse question, namely what can be said for the curves that admit such representation, it is shown that they are exactly the curves that can be decomposed into finitely many convex arcs. This result suggests the name piecewise convex curves. In particular, the class of piecewise convex curves contains the convex curves being boundary sets of convex figures, therefore the results from the paper can be used as a tool for studying convex curves.