  We consider embedded hypersurfaces evolving by fully nonlinear flows in which
the normal speed of motion is a homogeneous degree one, concave or convex
function of the principal curvatures, and prove a non-collapsing estimate:
Precisely, the function which gives the curvature of the largest interior
sphere touching the hypersurface at each point is a subsolution of the
linearized flow equation if the speed is concave. If the speed is convex then
there is an analogous statement for exterior spheres. In particular, if the
hypersurface moves with positive speed and the speed is concave in the
principal curvatures, then the curvature of the largest touching interior
sphere is bounded by a multiple of the speed as long as the solution exists.
The proof uses a maximum principle applied to a function of two points on the
evolving hypersurface. We illustrate the techniques required for dealing with
such functions in a proof of the known containment principle for flows of
hypersurfaces.
