Corrections to a paper of Allard and Almgren on the uniqueness of tangent cones

The paper referred to in the title is Allard/Almgren 1981 [AA81]. Several months ago Francesco Maggi emailed me saying that the inequalities 5.3(4),(5) of \cite{AA81} were wrong. In fact, as he pointed out, their incorrectness is immediately apparent if one takes $Z=0$ there. Maggi and his coauthor wanted to use these inequalities in their paper \cite{MN}. They were able to obtain a version of these inequalities which suffice for the carrying out the work in \cite{MN}. I started writing this paper in order to provide a version of these inequalities as needed in [AA81]. In thinking about this material I began to realize there were other problems with the paper. As a result I ended up {\em completely rewriting 5.1-5.4 on pages 243-248 of \cite{AA81}}; this rewrite is the contents of this paper. In addition to many annoying misprints many of the needed definitions and proofs in 5.1-5.4 are incomplete or absent. For example, it is not said where $z$ in 5.1(2) comes from; this omission completely surprised me since I remember doing a lot of work to come up with $z$ when \cite{AA81} was being written. Also, much of the necessary material about the reach of a submanifold as in \cite{FE2} is not provided in \cite{AA81}. The table of contents can serve as an index. In particular one sees there where the constants $\epsilon_1$ through $\epsilon_6$ are introduced. This material is extremely technical. One way to navigate this paper would be to start looking at Proposition 9.5 and work backwards.