On the right lower circular density of two-dimensional Jordan curves in the plane

Extract

We use J(a, b) to denote a Jordan curve of positive two-dimensional measure in the plane, with end-points a and b. If υ is a point of J(a, b), we define the right lower arc density at υ by

where J( υ, υ′) is the largest arc, whose left-end point is υ, which is contained in the disc c(υ, r).