A Bound on Local Minima of Arrangements that implies the Upper Bound Theorem

Clarkson Local Minima of Arrangements 6 Next to show that if v has outdegree i then v 0 is an i-minimum. Since v 0 j < 0 if and only if v 0 is below h j , we need to show that a coordinate v 0 j 6 = 0 corresponds to an oriented edge (v; q) where wv ? wq = w(v ? q) has the same sign as v 0 j. Suppose (v; q) is an edge of P. Then ^ Av = ^ b ^ Aq, with one strict inequality a j v = b j > a j q, and with equality for the other rows of ^ A. This implies that w(v ? q) = v 0 A(v ? q) = v 0 j a j (v ? q), and since a j (v ? q) > 0, v 0 j and w(v ? q) have the same sign. We have the Upper Bound Theorem, missing the proof that the given bound is tight for dual neighborly polytopes.