A New Approach to the Expansion of Positivity Set of Non-Negative Solutions toCertain Singular Parabolic Partial Differential Equations

{Let u be a non-negative solution to a singular parabolic equation of p-Laplacian type (1<p<2) or porous-medium type (0<m<1). If u is bounded below on a ball $B_\rho$ by a positive number M, for times comparable to $\rho$ and M, then it is bounded below by $\sigma M$, for some $\sigma\in(0,1)$, on a larger ball, say $B_{2\rho}$ for comparable times. This fact, stated quantitatively in Proposition Theorem1.1, is referred to as the ``spreading of positivity'' of solutions of such singular equations, and is at the heart of any form of Harnack inequality. The proof of such a ``spreading of positivity'' effect, first given in a paper by Chen and DiBenedetto, is rather involved and not intuitive. Here we give a new proof which is more direct being based on geometrical ideas.