We show the linear convergence of a simple first-order algorithm
for the minimum-volume enclosing ellipsoid problem and its dual,
the D-optimal design problem of statistics.
Using similar techniques we show the linear convergence of the Frank-Wolfe
algorithm with away steps applied to the simplex, under conditions
different from those of Gu\'{e}lat and Marcotte. Computational tests
confirm the attractive features of this method.