[dead]
The Shape of Inequalities
21–30 of 31 posts
Re: The Shape of Inequalities
#22Oh, these are really nice Andrei! Thanks for posting them.
Re: The Shape of Inequalities
#23The first chart is super confusing. The OP line is changing size as the circles move, yet (a-b)/2 is a constant.
Re: The Shape of Inequalities
#24My favorite bit of trivia is related to the following game: Start with 2 numbers, a and b and calculate HM and GM Now you have 2 numbers again, so you can play the game again with the new values Every step brings the results together, one from above, the other from below, sandwiching the value in the limit. That value is called Geometric-Harmonic Mean This works for all 3 pairs of means (HM-GM, GM-AM, HM-AM). The fun…
Also fun: The Arithmetic-Geometric Mean can be used to calculate Pi! (Most usefully, the AGM of 1 and sqrt(1/2).)
Re: The Shape of Inequalities
#25The book "When Less is More: Visualizing Basic Inequalities" by Claudi Alsina might also be of interest.
Re: The Shape of Inequalities
#26My favorite geometric proof of an inequality is the one I read on Terry Tao's blog. Interestingly, it's not presented as a geometric proof, but it is very much one: if you have two vectors x, y, you just shrink the longer one and grow the shorter one until they reach the same size, without changing the LHS and the RHS of the inequality. Then you expand the norms of ||x - y||^2>=0 and ||x + y||^2>=0 and see -||x||^2 - ||y||^2 <= ||x||^2 + ||y||^2, and since ||x||=||y|| you get the result.
Re: The Shape of Inequalities
#27The first chart is super confusing. The OP line is changing size as the circles move, yet (a-b)/2 is a constant.
I've fixed that.
Re: The Shape of Inequalities
#28The animated visuals are very cool, but I desperately want to turn them off in order to understand what they depict and reason about it geometrically. A pause button would be greatly appreciated.
Fixed.
Re: The Shape of Inequalities
#29The AM/GM inequality is why the world switched from (what in the UK is called) the Retail Price Index (AM) to the Consumer Price Index (GM).
Re: The Shape of Inequalities
#30My favorite bit of trivia is related to the following game: Start with 2 numbers, a and b and calculate HM and GM Now you have 2 numbers again, so you can play the game again with the new values Every step brings the results together, one from above, the other from below, sandwiching the value in the limit. That value is called Geometric-Harmonic Mean This works for all 3 pairs of means (HM-GM, GM-AM, HM-AM). The fun…
Also fun: The Arithmetic-Geometric Mean can be used to calculate Pi! (Most usefully, the AGM of 1 and sqrt(1/2).)
I was confused by this because pi is not between 1 and sqrt(1/2). But I found https://www.cs.miami.edu/home/burt/manuscripts/gaussagm/agma... and it clarifies that this AGM is used in the course of approximating pi not that it’s sufficient on its own to do so.