  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.
