Local corner cutting and the smoothness of the limiting curve (Q917213)

The author has proved earlier [ibid. 4, 125-131 (1987; Zbl 0637.41014)] that corner cutting of any kind of polygonal lines always leads to convergence to a Lipschitz continuous curve. Here he proves that ``local'' corner cutting (i.e., every cutting operation involves only one previous corner) in which the inner corner angles tend to \(\pi\) will lead to a \(C^ 1\) curve in the limit. The condition of eventual flatness is not necessary for local cuttings and not sufficient for nonlocal cuttings.