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Objects and object complexes in 3D, as well as those in 2D, have many possible representations. Among them skeletal representations have special advantages and some limitations. For the special form of skeletal representation called “s-reps,” these advantages include strong suitability for representing slabular object populations and statistical applications on these populations. Accomplishing these statistical applications is best if one recognizes that s-reps live on a curved shape space. Here we will lay out the definition of s-reps, their advantages and limitations, their mathematical properties, methods for fitting s-reps to single- and multi-object boundaries, methods for measuring the statistics of these object and multi-object representations, and examples of such applications involving statistics. While the basic theory, ideas, and programs for the methods are described in this paper and while many applications with evaluations have been produced, there remain many interesting open opportunities for research on comparisons to other shape representations, new areas of application and further methodological developments, many of which are explicitly discussed here. 

We frame the collision avoidance problem of multi-agent autonomous vehicle systems into an online convex optimization problem of minimizing certain aggregate cost over the time horizon. We then propose a distributed real-time collision avoidance algorithm based on the online gradient algorithm for solving the resulting online convex optimization problem. We characterize the performance of the algorithm with respect to a static offline optimization, and show that, by choosing proper stepsizes, the upper bound on the performance gap scales sublinearly in time. The numerical experiment shows that the proposed algorithm can achieve better collision avoidance performance than the existing Optimal Reciprocal Collision Avoidance (ORCA) algorithm, due to less aggressive velocity updates that can better prevent the collision in the long run.