The Average Directional Distance To The Boundary Of A Ball Or Disk

For a point P in the interior of the unit disk and a given direction, one can compute the directional distance from P to the boundary of the disk in the prescribed direction. We compute the average directional distance over all points and all directions. The analogous computation is then carried out for any n-dimensional ball. The average distance depends on the dimension n and produces a sequence which is shown to be a scale of the Wallis integral sequence. This work was motivated by research on solar energy collection devices.