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Every admissible linear decision boundary for the two-class multivariate normal recognition problem is known to be a hyperplane that is tangent to two tangent ellipsoids at their point of tangency. The ellipsoids are equiprobability surfaces for the distributions describing the classes. In this correspondence, the locus of tangent points is parameterized in a manner similar to that of Clunies-Ross and Riffenburgh. Anderson and Bahadur's work2 is then used to indicate which points on the locus give rise to admissible linear decision boundaries. A simple geometric proof is given for the characterization of admissible linear decision boundaries as tangent hyperplanes.