This blog post is entirely based on the start of this blog post by Terry Tao. I highly recommend reading the post. It gives an interesting insight into how Terry sometimes thinks about proving inequalities. He also gives a number of cool and more substantial examples.
The main idea in the blog post is that Terry likes to do “arbitrage” on an inequality to improve it. By starting with a weak inequality he exploits the symmetry of the environment he is working in to get better and better inequalities. He first illustrates this with a proof of the Cauchy-Schwarz inequality. The proof given is really nifty and much more memorable than previous proofs I’ve seen. I felt that just had to write it up and share it.
Let be an inner product space. The Cauchy-Schwarz inequality states that for all , . It’s an important result that leads, among other things, to a proof that satisfies the triangle inequality. There are many proofs of the Cauchy-Schwarz inequality but here is the one Terry presents.
Since is positive definite we have . Now using the fact that is additive in each coordinate we have
Since , we can rearrange the above expression to get the inequality
And now it is time to exploit the symmetry of the above expression and turn this inequality into the Cauchy-Schwarz inequality. The above inequality is worse than the Cauchy Schwarz inequality for two reasons. Firstly, unless is a positive real number, the left hand side is smaller than . Secondly, unless , the right hand side is larger than the quantity that we want. Indeed we want the geometric mean of and whereas we currently have the arithmetic mean on the right.
Note that the right hand side is invariant under the symmetry for any real number . Thus choose to be the negative of the argument of . This turns the left hand side into while the right hand side remains invariant. Thus we have done our first bit of arbitrage and now have the improved inequality
We now turn our attention to the right hand side and observe that the left hand side is invariant under the map for any . Thus by choosing we can minimize the right hand side. A little bit of calculus shows that the best choice is (this is valid provided that , the case when or is easy since we would then have ). If we substitute in this optimal value of , the right hand side of the above inequality becomes
Thus we have turned our weak starting inequality into the Cauchy-Schwarz inequality! Again I recommend reading Terry’s original post to see many more examples of this sort of arbitrage and symmetry exploitation.