<p><b>Abstract</b>—The problem of computing the convex hull of a set of <it>n</it> sorted points in the plane is one of the fundamental tasks in image processing, pattern recognition, cellular network design, and robotics, among many others. Somewhat surprisingly, in spite of a great deal of effort, the best previously known algorithm to solve this problem on a reconfigurable mesh of size <tmath>$\sqrt n\times \sqrt n$</tmath> was running in <it>O</it>(log<super>2</super><it>n</it>) time. It was open for more than ten years to obtain an algorithm for this important problem running in sublogarithmic time. Our main contribution is to provide the first breakthrough: We propose an almost optimal convex hull algorithm running in <it>O</it>((log log <it>n</it>)<super>2</super>) time on a reconfigurable mesh of size <tmath>$\sqrt n\times \sqrt n.$</tmath> With slight modifications, this algorithm can be implemented to run in <it>O</it>((log log <it>n</it>)<super>2</super>) time on a reconfigurable mesh of size <tmath>${\textstyle{{\sqrt n} \over {{\rm log\,log}\,n}}}\times {\textstyle{{\sqrt n} \over {{\rm log \,log}\,n}}}.$</tmath> Clearly, the latter algorithm is work-optimal. We also show that any algorithm that computes the convex hull of a set of <it>n</it> sorted points on an <it>n</it>-processor reconfigurable mesh must take Ω(log log <it>n</it>) time. Our result opens the door to an entire slew of efficient convex-hull-based algorithms on reconfigurable meshes.</p>