  We study the problem of covering a given set of $n$ points in a high,
$d$-dimensional space by the minimum enclosing polytope of a given arbitrary
shape. We present algorithms that work for a large family of shapes, provided
either only translations and no rotations are allowed, or only rotation about a
fixed point is allowed; that is, one is allowed to only scale and translate a
given shape, or scale and rotate the shape around a fixed point. Our algorithms
start with a polytope guessed to be of optimal size and iteratively moves it
based on a greedy principle: simply move the current polytope directly towards
any outside point till it touches the surface. For computing the minimum
enclosing ball, this gives a simple greedy algorithm with running time
$O(nd/\eps)$ producing a ball of radius $1+\eps$ times the optimal. This simple
principle generalizes to arbitrary convex shape when only translations are
allowed, requiring at most $O(1/\eps^2)$ iterations. Our algorithm implies that
{\em core-sets} of size $O(1/\eps^2)$ exist not only for minimum enclosing ball
but also for any convex shape with a fixed orientation. A {\em Core-Set} is a
small subset of $poly(1/\eps)$ points whose minimum enclosing polytope is
almost as large as that of the original points. Although we are unable to
combine our techniques for translations and rotations for general shapes, for
the min-cylinder problem, we give an algorithm similar to the one in
\cite{HV03}, but with an improved running time of $2^{O(\frac{1}{\eps^2}\log
\frac{1}{\eps})} nd$.
