The reverse Poincaré inequality
As explained in Remark 23.8, the Lipschitz approximation theorem reduces the regularity problem by showing that, by perimeter minimality, the smallness of the excess at a given scale implies the smallness of the excess at every smaller scale. To prove this, in Chapter 25, we shall need a reverse height bound, in which the excess is controlled through a sort of L2-height. Precisely, we introduce the cylindrical flatness of a set of locally finite perimeter at with respect to at scale r > 0, as
The flatness ...