Convexity : an analytic viewpoint Convexity of sets and functions are extremely simple notions to define so it may be somewhat surprising the depth and breadth of ideas that these notions give rise to. It turns out that convexity is central to a vast number of applied areas including Statistical Mechanics Thermodynamics Mathematical Economics and Statistics and that many inequalities including Hl̲ders and Minkowskis inequalities are related to convexity. An introductory chapter (1) includes a study of regularity properties of convex functions some inequalities (Hl̲der Minkowski and Jensen) the Hahn Banach theorem as a statement about extending tangents to convex functions and the introduction of two constructions that will play major roles later in this book: the Minkowski gauge of a convex set and the Legendre transform of a function