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Farkas proved his famous result (which, nowadays, is fundamental in optimization theory) in 1902 and called it Grundsatz der einfachen Ungleichung which may be translated as fundamental theorem of simple inequalities (where simple means linear, I know that Grundsatz can also be translated as principle, but reading Farkas, I strongly believe that fundamental theorem fits better).

Question. Who was the first to call his fundamental theorem the Farkas' lemma?

I have edited the question because of the comments which concern the former formulation of a relegation.

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    $\begingroup$ I don't know, but "relegation" makes it sound like a demotion. It's not! Rota said somewhere that what mathematicians secretly want is a lemma named after them. Lucky Yoneda, lucky Nakayama. Results that, while they may be simple to prove, are profound and important by virtue of the fact they get used everyday by everyone in the field. I don't think it's outrageous to think of Farkas's lemma that way. $\endgroup$
    – Todd Trimble
    Commented May 12, 2022 at 15:27
  • $\begingroup$ FWIW: I think "lemma" is also a much more beautiful name for a child/pet than "theorem". $\endgroup$
    – pinaki
    Commented May 12, 2022 at 16:06
  • $\begingroup$ A lemma is not inferior to a theorem, it just means that it it an intermediary results towards another objective. It does not mean that lemmas are not important. Also lemmas are not necessarily easier to prove than theorems. I have a couple of non-elementary lemmas which are fundamental. The Kalman-Yakubovich-Popov Lemma is one of them, for instance. $\endgroup$
    – KBS
    Commented May 21, 2022 at 22:01
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    $\begingroup$ Lemmas do the work in Mathematics. Theorems, like Management, just take the credit. $\endgroup$
    – Paul Taylor
    Commented May 23, 2022 at 7:18

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This is the earliest reference I have located:

Minkowski-Farkas Lemma in Banach Spaces, L. Hurwicz (1952).

The same result was also referred to as the Minkowski-Farkas-Weyl theorem in the 1950's, for example in

The strong Minkowski-Farkas-Weyl theorem for vector spaces over ordered fields, A. Charnes and W.W. Cooper (1958).

The three authors are associated with this result in view of the following contributions:

  • H. Minkowski, Geometrie der Zahlen (Leipzig, 1896).
  • J. Farkas, Über die Theorie der einfachen Ungleichungen, Journal für die reine und angewandte Mathematik, 124, 1-27 (1902).
  • H. Weyl, Elementare Theorie der konvexen Polyeder, Comm. Helvet., 7, 290-306 (1935).
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