In the proof of Corollary 3.2; I believe \bm{y} = [-\langle\bm{v},\bm{z}\rangle, x_1,\dots,x_m] should, in fact, be defined as \bm{y} = [-\langle\bm{v},\bm{z}\rangle, z_1,\dots,z_m].
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In the proof of Corollary 3.2; I believe \bm{y} = [-\langle\bm{v},\bm{z}\rangle, x_1,\dots,x_m] should, in fact, be defined as \bm{y} = [-\langle\bm{v},\bm{z}\rangle, z_1,\dots,z_m].