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THE CAUCHY – SCHWARZ MASTER CLASS - PART 2

Cauchy’s Second Inequality: The AM-GM Bound Thảo luận ban đầu của chúng tôi về bất đẳng thức Cauchy của lình xình về việc áp dụng của bất đẳng thức cơ bản biến thực xy ≤ y2 x2 + 2 2 với mọi x, y ∈ R, (2.1) và một cách đúng đắn có thể tự hỏi làm thế nào nhiều giá trị có thể được rút ra từ một ràng buộc mà xuất phát từ các quan sát tầm thường mà (x - y) 2 ≥ 0. | 2 Cauchy s Second Inequality The AM-GM Bound Our initial discussion of Cauchy s inequality pivoted on the application of the elementary real variable inequality x2 y2 _ xy 2 2 for all x y G R 2-1 and one may rightly wonder how so much value can be drawn from a bound which comes from the trivial observation that x y 2 0. Is it possible that the humble bound 2.1 has a deeper physical or geometric interpretation that might reveal the reason for its effectiveness For nonnegative x and y the direct term-by-term interpretation of the inequality 2.1 simply says that the area of the rectangle with sides x and y is never greater than the average of the areas of the two squares with sides x and y and although this interpretation is modestly interesting one can do much better with just a small change. If we first replace x and y by their square roots then the bound 2.1 gives us 4yxy 2x 2y for all nonnegative x y 2.2 and this inequality has a much richer interpretation. Specifically suppose we consider the set of all rectangles with area A and side lengths x and y. Since A xy the inequality 2.2 tells us that a square with sides of length s ựxỹ must have the smallest perimeter among all rectangles with area A. Equivalently the inequality tells us that among all rectangles with perimeter p the square with side s p 4 alone attains the maximal area. Thus the inequality 2.2 is nothing less than a rectangular version of the famous isoperimetric property of the circle which says that among all planar regions with perimeter p the circle of circumference p has the largest area. We now see more clearly why xy x2 2 y2 2 might be 19 20 The AM-GM Inequality powerful it is part of that great stream of results that links symmetry and optimality. From Squares to n-CuBEs One advantage that comes from the isoperimetric interpretation of the bound xy x y 2 is the boost that it provides to our intuition. Human beings are almost hardwired with a feeling for geometrical truths and one can easily .