  We give near-optimal algorithms for computing an ellipsoidal rounding of a
convex polytope whose vertices are given in a stream. The approximation factor
is linear in the dimension (as in John's theorem) and only loses an excess
logarithmic factor in the aspect ratio of the polytope. Our algorithms are
nearly optimal in two senses: first, their runtimes nearly match those of the
most efficient known algorithms for the offline version of the problem. Second,
their approximation factors nearly match a lower bound we show against a
natural class of geometric streaming algorithms. In contrast to existing works
in the streaming setting that compute ellipsoidal roundings only for centrally
symmetric convex polytopes, our algorithms apply to general convex polytopes.
We also show how to use our algorithms to construct coresets from a stream of
points that approximately preserve both the ellipsoidal rounding and the convex
hull of the original set of points.
