Much of the past work on asynchronous approximate Byzantine consensus has assumed scalar inputs at the nodes [3, 7]. Recent work has yielded approximate Byzantine consensus algorithms for the case when the input at each node is a d-dimensional vector, and the nodes must reach consensus on a vector in the convex hull of the input vectors at the fault-free nodes [8, 12]. The d-dimensional vectors can be equivalently viewed as points in the d-dimensional Euclidean space. Thus, the algorithms in [8, 12] require the fault-free nodes to decide on a point in the d-dimensional space. In this paper, we generalize the problem to allow the decision to be a convex polytope in the d-dimensional space, such that the decided polytope is within the convex hull of the input vectors at the fault-free nodes. We name this problem as Byzantine convex consensus (BCC), and present an asynchronous approximate BCC algorithm with optimal fault tolerance. Ideally, the goal here is to agree on a convex polytope that is as large as possible. While we do not claim that our algorithm satisfies this goal, we show a bound on the output convex polytope chosen by our algorithm.
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