Generalized Self-Approaching Curves

We consider all planar oriented curves that have the following property depending on a xed angle '. For each point B on the curve, the rest of the curve lies inside a wedge of angle ' with apex in B. This property restrains the curve's meandering, and for ' 2 this means that a point running along the curve always gets closer to all points on the remaining part. For all ' < , we provide an upper bound c(') for the length of such a curve, divided by the distance between its endpoints, and prove this bound to be tight. A main step is in proving that the curve's length cannot exceed the perimeter of its convex hull, divided by 1 + cos'.