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Submitted by R.A. Brualdi AMS classification: 15A45 15A03 15A60 65F35 90C47 Keywords: Matrix inequalities Distance to lower rank Functional analysis in matrix theory Condition number Four essentially different interpretations of a lower bound for linear operators are shown to be equivalent for matrices (involving inequalities, convex sets, minimax problems, and quotient spaces). Properties stated by von Neumann in a restricted case are satisfied by the lower bound. Applications are made to rankdoi:10.2172/836372 fatcat:k2p3uqq7pjegbdi3go6w36o7oa