Inequalities of convexity

Nicolas Bourbaki

Chapter Chapter I in Elements of Mathematics, 2004, pp 1-7 from Springer

Abstract: Abstract Let X be a set; in the vector space RX of all finite numerical functions1 defined on X, let P be the set of all positive real-valued functions on X. On the other hand, let M be a numerical function2, finite or not, with values ≥ 0, defined on P, such that: 1° M (0) = 0, and M is positively homogeneous, that is, M (λf) = λM (f) for every real number λ > 0.3 2° M is increasing in P, in other words the relation f ≤ g implies M (f) ≤ M (g). 3° M is convex in P, in other words (TVS, II, §2, No. 8) satisfies the relation M (f +g) ≤ M (f) + M (g).

Date: 2004
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DOI: 10.1007/978-3-642-59312-3_1

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