  Approximating convex bodies is a fundamental question in geometry and has a
wide variety of applications. Given a convex body $K$ of diameter $\Delta$ in
$\mathbb{R}^d$ for fixed $d$, the objective is to minimize the number of
vertices (alternatively, the number of facets) of an approximating polytope for
a given Hausdorff error $\varepsilon$. The best known uniform bound, due to
Dudley (1974), shows that $O((\Delta/\varepsilon)^{(d-1)/2})$ facets suffice.
While this bound is optimal in the case of a Euclidean ball, it is far from
optimal for ``skinny'' convex bodies.
  A natural way to characterize a convex object's skinniness is in terms of its
relationship to the Euclidean ball. Given a convex body $K$, define its surface
diameter $\Delta_{d-1}$ to be the diameter of a Euclidean ball of the same
surface area as $K$. It follows from generalizations of the isoperimetric
inequality that $\Delta \geq \Delta_{d-1}$.
  We show that, under the assumption that the width of the body in any
direction is at least $\varepsilon$, it is possible to approximate a convex
body using $O((\Delta_{d-1}/\varepsilon)^{(d-1)/2})$ facets. This bound is
never worse than the previous bound and may be significantly better for skinny
bodies. The bound is tight, in the sense that for any value of $\Delta_{d-1}$,
there exist convex bodies that, up to constant factors, require this many
facets.
  The improvement arises from a novel approach to sampling points on the
boundary of a convex body. We employ a classical concept from convexity, called
Macbeath regions. We demonstrate that Macbeath regions in $K$ and $K$'s polar
behave much like polar pairs. We then apply known results on the Mahler volume
to bound their number.
