TABLE OF CONTENTS In this Appendix, we prove Theorems 1 and 2 of Section 4.2.4, and justify the need for the interior in Theorem 2.
First, suppose that b ? Q is a strict normal. If there is a ? ? 
with ? ? w, then in particular ? 
w, that ( ? ? w) T r ? 0 for all r ? Q. But this contradicts the hypothesis.
Second, suppose that b ? 
is a normal that is, that b T( ? ? w) ? 0 for all ? ? . Suppose there is a ? ? with ? 
w (that is, ( ? ? w) T r ? 0 for all r ? Q) then ( ? ?w) T b = 0. Since b ? , for any s ? R, b + ?s ? Q for ? small. Then ( ? ? w) T( b + ?s) ? 0, which implies that ( v ? w) T s = 0 for all s ? R, so ? ? w ? R ?. But then it is impossible that ( v ? w) T r > 0 for any r ? Q, so v ? w.
Simple examples show that the strictness conditions are necessary.
