Once you've taken the logarithm of both sides, the proof that the logarithm of the average is larger than the average of the logarithms seems pretty easy.
I think there is an interesting analogy there to a famous picture from nuclear physics, the curve of binding energy per nucleon in an atomic nucleus -- the graph that shows that fusion of nuclei below iron or fission of nuclei above iron can yield energy, and that iron nuclei are the most stable. ("Everything wants to be iron.") If you look at the region of that curve which shows where fusion occurs, you see that the average (arithmetic mean) energy is greater when you move to the right, in the increasing direction, toward iron. The curve is everywhere concave downward, so that in its increasing region, between hydrogen and iron, the average energy of the sum of nucleons (into a single nucleus) is always greater than the sum of the average energies of nucleons (in separate nuclei).
Where the curve is decreasing (but still concave downward), between iron and infinity, the average energy of the sum of nucleons is always less than the sum of their average energies in separate nuclei.
So, with the arithmetic and geometric mean inequality, after you've taken the logarithm of both sides, the problem is quite similar to explaining how the curve of binding energy shows which nuclear reactions will release energy as a result of changes in mass deficit.
Sorry if that wasn't completely clear; it's very difficult for me to talk about the curve of binding energy without having one handy to point at.
In other news
I had a great dinner with a bookseller couple in Berkeley.
On Friday, my friend Anirvan took me to see a pair of documentaries at a movie theater in Berkeley. Called "Showdown in Seattle" and "Breaking the Bank", they concerned two major U.S. demonstrations of the past year (the WTO protest in Seattle and the World Bank protest in Washington, DC). The two of these were probably the most distressing thing I've seen in years, and definitely "Showdown in Seattle" was the most violent movie I've seen in a long time.
I could probably write about these two documentaries for pages, and I probably will, but maybe in a letter rather than on Advogato.
After that, and once I could breathe again (I discovered that watching pictures of people getting teargassed does interesting things to my autonomic nervous system), we went to a post-election party at Sumana's place, and it was very geeky and a lot of fun. I got to cite Martin Gardner on voting paradoxes, and we played Hangman, and various other things.
Hi, Anirvan, Sumana, Michelle, Nathaniel. Did I miss anybody?
I stayed over in Berkeley and got up extremely early to make a 3.5-hour trip from Berkeley to Santa Rosa by way of San Francisco. I'm still yawning and falling asleep intermittently.
I was in Santa Rosa because I was invited to attend a technology planning meeting for Sonoma Academy, a new independent high school which will open next year. (Non-U.S. readers: an independent school is a private, i.e. non-tax-supported, competitive-admission, tuition-based school which isn't affiliated with a larger organization like a religious denomination.)
The meeting was pretty cool. We covered a lot of ground, and I met some nice people. It's amazing to be in a situation where people are talking about computer technology and they aren't even considering using free software -- in fact, they haven't even heard of free software! Most of the people at the meeting assumed that the school would be using Microsoft Windows for all its computing.
All the other computer people I've met in recent memory were free software deveopers, or free software advocates, or the like. I have pretty much entirely stopped interacting with technologists outside the free software world. (The notable exception was Dave Winer; lilo and jpick and I met him at the Moscone Center, and that was interesting.) So it's kind of a shock to have this reality check, like, not only do these people not assume that they will use Linux or another free PC Unix, but they assume instead that they will use Windows 2000 or another proprietary Microsoft operating system.
I guess it's like spending a couple of years in a somewhat insular religious community and then venturing outside momentarily.
If I have any further influence in the concrete decision-making for Sonoma Academy's IT policy, I'll try to get them to consider Linux for all their new servers and maybe for some lab machines, too. But I think I made some useful general points completely outside of the free software issue, on the subject of the role of information technology in high school education. So even if they go with Windows everywhere, I hope my general points will have had an effect.
Sonoma County is pretty beautiful, like much of California, but I was either asleep on a bus or engrossed in conversation about LBL each time I was passing through, so I didn't actually get to see much of it.
I think I had better become a teacher or something.
It's gotten cold recently, in San Francisco, Berkeley, and Santa Rosa alike. I guess I'll have to start wearing socks and a jacket or something.