Motzkin’s theorem of the alternative: a continuous-time generalization

Abstract

We revisit Zalmai’s theorem, which is a partial generalization of Motzkin’s theorem of the alternative in the continuous-time setting. In particular, we provide two simple examples demonstrating that its existing proof is incorrect, and we demonstrate that a suitably modified variant of Zalmai’s theorem, concerned with the inconsistency of systems of convex inequalities and affine equalities, can be verified. We also derive two generalized variants of Motzkin’s theorem of the alternative in the continuous-time setting.