  It is known that the gradient descent algorithm converges linearly when
applied to a strongly convex function with Lipschitz gradient. In this case the
algorithm's rate of convergence is determined by the condition number of the
function. In a similar vein, it has been shown that a variant of the
Frank-Wolfe algorithm with away steps converges linearly when applied to a
strongly convex function with Lipschitz gradient over a polytope. In a nice
extension of the unconstrained case, the algorithm's rate of convergence is
determined by the product of the condition number of the function and a certain
condition number of the polytope.
  We shed new light into the latter type of polytope conditioning. In
particular, we show that previous and seemingly different approaches to define
a suitable condition measure for the polytope are essentially equivalent to
each other. Perhaps more interesting, they can all be unified via a parameter
of the polytope that formalizes a key premise linked to the algorithm's linear
convergence. We also give new insight into the linear convergence property. For
a convex quadratic objective, we show that the rate of convergence is
determined by a condition number of a suitably scaled polytope.
