On the simultaneous solution of a certain system of linear inequalities
George J. Minty · Proceedings of the American Mathematical Society · 1962
The author recently proved the following theorem: THEOREM 1. Let X be a Hilbert space, with real or complex scalars and inner product (x, y). Let xi, * *, xa and yi, * ym be given such that (1) Re(x xi, y yi) 2! 0 (;i, = 1, ... ) and let x be any point of X. Then there exists a point y such that (2) Re(xi -x, yiy) _ 0i=1 ) The proof was patterned after Schoenberg's [3] proof of Kirszbraun's theorem. B. Griinbaum [2] has generalized my and Schoenberg's proofs to obtain a theorem which incorporates Theorem 1 and Kirszbraun's theorem, and J. G. Wendel has contributed a neater proof of Theorem 1. With Professor Wendel's permission, I reproduce his proof: LEMMA. Let XE be En, with the usual (real) scalars and inner product, and let xi, , x..; yi, * * , ym be given such that (11) (Xi Xj) y-yj) _ 0 (i= *Xm) Then there exists a point y such that (2') (x;, y) < (xi, yi) (i = 1, * , m). PROOF OF THE LEMMA. Let A be the matrix whose ith row is xi, and let b be the column-vector whose ith element is (xi, yi). Then (2') is equivalent to Ay_ b. If there is no solution for y, then by Stiemke's Theorem [1, Theorem 2.7] there exists a row-vector n < 0 such that (3a, 3b) qA = 0 and qb = 1. Suppose this to be the case. Then (3a) implies Ei nixi= 0, hence for each j, (4) E ni(xi, yj) = 0.