Tracking problem: differentiation of the visibility function (Q416968)

Assume that the positions of an observer and an object are known to be in certain closed balls (possibly moving in time \(t\)). Given an obstacle represented by a closed set the object is visible to the observer if the convex hull of the union of the balls does not intersect with the obstacle (for all \(t\)). The visibility function assigning (for every \(t\)) the largest radius of the ball around the object preserving visibility is defined. Certain differentiability properties are proven for this assignment. If the object moves from one position to another it has to do this on certain trajectories avoiding collision with the obstacle. If in addition it tries to minimize the radius which is observable around it at all positions on the trajectory, this is related to the visibility function. It is proven in a algorithmic manner that such a trajectory exists and is smooth.