My Math Forum geometry from simple inequality!

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 July 28th, 2010, 02:43 PM #1 Senior Member   Joined: Dec 2009 Posts: 150 Thanks: 0 geometry from simple inequality! From the Cauchy-Schwarz Master Class : An Introduction to the Art of Inequalities by J. Micheal Steele We start with the fact that a real number squared is at least zero so that $(x-y)^2 \,\geq\, 0$ and derive a property about squares and rectangles: $(x-y)^2 \geq 0 \,\, \Rightarrow \, \, 2xy \, \leq \, x^2 + y^2$ replacing the x and y with$\sqrt{x}$ and $\sqrt{y}$ and multiplying both sides by 2 we have that $4\sqrt{x}\sqrt{y} \,= \, 4\sqrt{xy} \, \leq \, 2\sqrt{x}^2 + 2\sqrt{y}^2 \, = \, 2x + 2y$ Interpreting xy as an area A = xy, we have that for a square the side length is s = \sqrt{A} and 4s = [perimeter of the square ] Thus we have shown that of all rectangles that have area A = xy, the square is the one with the smallest perimeter! (the other rectangles perimeters being the upper bound 2x + 2y). Isn't that cool, how the positivity of square numbers more or less directly implies that the square is the most efficient way to surround an area with a given perimeter. Symmetry and optimality are often two ways of describing the same situation. (Symmetric rectangle is [square] is the optimal one). The same inequality can be used to show that if two series $\sum^{\infty}_{k=1} a_k^2$ and $\sum^{\infty}_{k= 1} b_k^2$ converge then so does $\sum^{\infty}_{k=1} |a_k b_k|$ (show the same equality holds for each term in the series so also for the partial sums so also in the limit). A certain normalization of the series (so that the sum of the squares of the normalized a_k, b_k equals 1) converts the same inequality into Cauchy's inequality for series!! This goes to show that Cauchy's equality is very geometric, and that simple inequalities (indeed obvious ones) can sometimes pack a BIG punch if you play with them enough. Hope you enjoyed, and please post any similar examples of the obvious leading to the non-obvious, or inequalities and their geometric interpretation.

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