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Xn ) is convex in a region if and only if the Hessian matrix Hij = ∂2 ∂xi ∂xj 30 is positive definite everywhere in the region. Note that the Hessian is symmetric because of the symmetry of mixed partials, so this statement makes sense. Proof. The function f is convex if and only if its restriction to each line is convex, and the second derivative along a line through x in the direction of y is (up to a scale factor) just Hy · y evaluated at x. So f is convex if and only if Hy · y > 0 for all nonzero y, that is, if H is positive definite.

Rest assured that none of the following material is referenced anywhere else in the notes. We have a pretty good understanding now of the relationship between extremization and the derivative, for functions in a single variable. However, most inequalities in nature deal with functions in more than one variable, so it behooves us to extend the formalism of calculus to this setting. Note that whatever we allow the domain of a function to be, the range of our “functions” will always be the real numbers.

2] G. 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.

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A is less than B by Kedlaya K.S.

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