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lower bound; linear proofs; Helly Theorem

References:

[1] M. O. Rabin: **Proving simultaneous positivity of linear forms**. JCSS 6 : 639-650 (1972). MR 0451858 | Zbl 0274.68022

[2] Joseph Stoer, Christoph Witzgall: **Convexity and Optimization in Finite Dimensions I**. Springer-Verlag, Berlin-Heidelberg-New York, J 970.

[3] Ky Fan: **On systems of linear inequalities**. in 'Linear Inequalities and Related Systems' (H. W. Kuhn and A. W. Tucker, eds.). Princeton Univ. Press, 1956. MR 0087901 | Zbl 0072.37602

[4] David G. Luenberger: **Introduction to Linear and Nonlinear Programming**. Addison-Wesley Publishing Company, 1973.

[5] J. W. Jaromczyk: **An extension of Rabin's complete proof concept**. MFCS 1981, Lecture Notes in Computer Science 118, Springer-Verlag 1981, 321 - 326. MR 0652765 | Zbl 0471.68026