Arc length associated with generalized distance functions
The authors consider generalized distance functions \(d\) which do not satisfy the usual properties except that \(d(x, x)= 0\) for any \(x\). An arc length of rectifiable curves is introduced in the usual manner. Under an additional assumption on the one-sided directional derivative an explicit expression of the arc length is obtained in terms of an integral. Finally an example is given for a non-positive definite and asymmetric distance function.
scientific article; zbMATH DE number 5757937