ON TUCKER ’ S KEY THEOREM

A new proof of a (slightly extended) geometric version of Tucker’s fundamental result is given.

A classical result of A. W. Tucker (9) states that the dual systems In this note we suggest a new proof of a (slightly extended) geometric version of this fundamental result, which was observed, e.g.(6), to be a key to duality theory.
We use L and S to denote convex cones in Cn, i.e., subsets of the n-dimensional unitarian space which are closed under addition and under multiplication by a nonnegative scalar.For a nonempty set T =_ Cn, T* denotes the closed convex cone {x e cn; Re(x,T) -> 0}.
We shall make use of the following identities: K* (cK)*, (2.1) satisfied by the convex cones K, and K 2. For these and other basic results on convex cones the reader is referred to (2) and (4).
Consider the following intersection.

I(L,S) S S*O (LOS)*O (-LnS)*)O (L-S).
The proof of the main result is based on the fact that this intersection consists only of the origin. LEMMA.I(L,S)---{0}.
PROOF.Let x I(L,S).Then x e L S and there exists an s e S such that x+s eL.Now, x e S -)x + s e S =)x + s e L S.
Still more, x e S* => Re(x,s) __> 0. Thus < 0, but this is possible only when x 0, which was to be proved The intersection SNS* is pointed. (It consists only of the origin if and only if S is a real subspace, e.g.(i), (5)).Thus (SS*)* is solid and the following theorem is meaningful.KEY THEOREM.If (i) L S is closed or (iia) L* + S* is closed and (lib) c(LNS) c L c S, then x LOS, v e (S-L)*, x + v e int(SnS*)*, (2.4)   is consistent.
The assumptions made in the theorem suggest two interrelated open problems: a) Is the theorem true without the assumptions?
b) For what convex cones L and S, both assumptions, (i) and (ii), do not hold?
In conclusion, we point out some special cases.
The real version of the theorem with S R is due to Epelman and Waksman (3).
Taking S to be polyhedral and L the null space of a matrix A,L* L + R(AH) and replacing v e R(AH)s * by AHy S* one gets the Key Theorem of Abrams and Ben-lsrael (i).As shown in (i), the theorem of Tucker is the real special case where S R. Its complex extension, due to Levinson (8) (i) < T e, e-vector of ones and S* T ACKNOWLEDGMENT.We wish to thank the referee for his remarks and suggestions.