  Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a
continuous one-parameter family of closed curves starting and ending at point
curves. Pull the sweepout tight by, in a continuous way, pulling each curve as
tight as possible yet preserving the sweepout. We show:
  Each curve in the tightened sweepout whose length is close to the length of
the longest curve in the sweepout must itself be close to a closed geodesic. In
particular, there are curves in the sweepout that are close to closed
geodesics.
  Finding closed geodesics on the 2-sphere by using sweepouts goes back to
Birkhoff in 1917.
  As an application, we bound from above, by a negative constant, the rate of
change of the width for a one-parameter family of convex hypersurfaces that
flows by mean curvature. The width is loosely speaking up to a constant the
square of the length of the shortest closed curve needed to ``pull over'' $M$.
This estimate is sharp and leads to a sharp estimate for the extinction time;
cf. [CM1], [CM2] where a similar bound for the rate of change for the two
dimensional width is shown for homotopy 3-spheres evolving by the Ricci flow
(see also Perelman).

Differential Geometry;Analysis of PDEs;Geometric Topology