This paper describes optimal algorithms for solving some fundamental geometric problems on a mesh connected parallel computer: minimum enclosing circle, detecting circular separability of two point sets, recognition of digital disks, finding the smallest and the largest separating circle of two sets of points, detecting linear separability of two sets, inclusion of points in a convex polygon, maximum gap problem and the diameter of a binary image. The problem of finding the diameter of a black/white picture is solved in general, for $k$-dimensional binary image stored one pixel per processor in a $k$-dimensional mesh connected computer, and for any monotone metric.