Convexity — The Third Pillar Có ba trụ cột lớn của lý thuyết của sự bất bình đẳng tích cực, monotonicity, và lồi. Các khái niệm tích cực và monotonicity là nội tại để các đối tượng mà họ phục vụ chúng tôi đều đặn mà không bao giờ kêu gọi sự chú ý cho chính mình, nhưng lồi là khác nhau. Lồi thể hiện một tác dụng lệnh thứ hai, và cho nó để cung cấp trợ giúp chúng tôi gần như luôn luôn cần phải thực hiện một số chế phẩm có chủ ý. Để bắt đầu, trước. | 6 Convexity The Third Pillar There are three great pillars of the theory of inequalities positivity monotonicity and convexity. The notions of positivity and monotonicity are so intrinsic to the subject that they serve us steadily without ever calling attention to themselves but convexity is different. Convexity expresses a second order effect and for it to provide assistance we almost always need to make some deliberate preparations. To begin we first recall that a function f a b R is said to be convex provided that for all x y G a b and all 0 p 1 one has f px 1 - p y pf x 1 - p f y . With nothing more than this definition and the intuition offered by the first frame of Figure we can set a challenge problem which creates a fundamental link between the notion of convexity and the theory of inequalities. Problem Jensen s Inequality Suppose that f a b R is a convex function and suppose that the nonnegative real numbers pj j 1 2 . . n satisfy pi p2 ---- pn 1. Show that for all Xj G a b j 1 2 . . n one has n n f ITpj j pjf xj . When n 2 we see that Jensen s inequality is nothing more than the definition of convexity so our instincts may suggest that we look for a proof by induction. Such an approach calls for one to relate averages of size n - 1 to averages of size n and this can be achieved several ways. 87 88 Convexity The Third Pillar Fig. . By definition a function f is convex provided that it satisfies the condition which is illustrated in frame A but a convex function may be characterized in several other ways. For example frame B illustrates that a function is convex if and only if its sequential secants have increasing slopes and frame C illustrates that a function is convex if and only if for each point p on its graph there is line through p that lies below the graph. None of these criteria requires that f be differentiable. One natural idea is simply to pull out the last summand and to renormalize the sum that is left behind. More .
