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By Kiran Kedlaya

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Bucharest 40 (1934), 155-160. [4] S. Rabinowitz, Index to Mathematical Problems 1980-1984, Mathpro Press, Westford (Massachusetts), 1992.

E. Littlewood, and G. P´olya, Inequalities (second edition), Cambridge University Press, Cambridge, 1951. [3] T. Popoviciu, Asupra mediilor aritmetice si medie geometrice (Romanian), Gaz. Mat. Bucharest 40 (1934), 155-160. [4] S. Rabinowitz, Index to Mathematical Problems 1980-1984, Mathpro Press, Westford (Massachusetts), 1992.

5 Convexity in several variables A function f defined on a convex subset of a vector space is said to be convex if for all x, y in the domain and t ∈ [0, 1], tf (x) + (1 − t)f (y) ≥ f (tx + (1 − t)y). Equivalently, f is convex if its restriction to any line is convex. Of course, we say f is concave if −f is convex. The analogue of the second derivative test for convexity is the Hessian criterion. A symmetric matrix M (that is, one with Mij = Mji for all i, j) is said to be positive definite if M x·x > 0 for all nonzero vectors x, or equivalently, if its eigenvalues are all real and positive.

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A < B by Kiran Kedlaya


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