Vol. 20, No. 1, 1967

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Extreme copositive quadratic forms. II

Leonard Daniel Baumert

Vol. 20 (1967), No. 1, 1–20

A real quadratic form Q = Q(x1,,xn) is called copositive if Q(xI,,xn) 0 whenever x1,,xn 0. If we associate each quadratic form Q = qijxixjqij = qji(i,j = 1,,n) with a point

             -         -
(q11,⋅⋅⋅ ,qnn,√2q12,⋅⋅⋅ ,√ 2q∞1,n)

of Euclidean n(n + 1)2 space, then the copositive forms constitute a closed convex cone in this space. We are concerned with the extreme points of this cone. That is, with those copositive quadratic forms Q for which Q = Q1 + Q2 (with Q1,Q2 copositive) implies Q1 = αQ,Q2 = (1 a)Q,0 a 1. In this paper we limit ourselves almost entirely to 5-variable forms and announce the discovery of an hitherto unknown class of extreme copositive quadratic forms in 5 variables. In view of the known extension process whereby extreme copositive quadratic forms in n variables may be used to generate extreme forms in nvariables for any n> n > 2, this new class of forms thus provides new extreme copositive forms in any number of variables n5.

Copositive quadratic forms arise in the theory of inequalities and also in the study of block designs. The paper of Diananda [2] provides the connection with inequalities while the paper of Hall and Newman [3] outlines the application of copositive quadratic forms to block designs.

Mathematical Subject Classification
Primary: 15.70
Received: 5 April 1965
Revised: 23 July 1965
Published: 1 January 1967
Leonard Daniel Baumert