  Much of the past work on asynchronous approximate Byzantine consensus has
assumed scalar inputs at the nodes [4, 8]. 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 [9, 13]. The
d-dimensional vectors can be equivalently viewed as points in the d-dimensional
Euclidean space. Thus, the algorithms in [9, 13] require the fault-free nodes
to decide on a point in the d-dimensional space.
  In our recent work [arXiv:/1307.1051], we proposed a generalization of the
consensus problem, namely Byzantine convex consensus (BCC), which allows 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 also presented an asynchronous approximate BCC algorithm.
  In this paper, we propose a new BCC algorithm with optimal fault-tolerance
that also agrees on a convex polytope that is as large as possible under
adversarial conditions. Our prior work [arXiv:/1307.1051] does not guarantee
the optimality of the output polytope.
