Kakutani is at rest

Shizuo Kakutani, the bane of many a graduate student in economics for his fixed point theorem, has died at the age of 92. Kakutani's theorem is difficult to explain but it is a generalization of Brouwer's fixed point theorem which has some elegant illustrations. Consider the following problem.

One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit. The monk ascended the path at varying rates of speed, stopping many times along the way to rest and to eat the dried fruit he carried with him. He reached the temple shortly before sunset. After several days of fasting and meditation he began his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way. His average speed descending was, of course, greater than his average climbing speed.

Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.

The statement to be proved is true but rather surprising. The monk, after all moves up and down the mountain at varying rates of speed and yet regardless of how he travels Brouwer's fixed point theorem tells us that there is a time at which the monk is at exactly the same elevation on the upward and downward trips. There is in fact an easy proof for this problem which I will put into the extension. Before getting there here are some more implications of the theorem in two and three dimensions.

Take two pieces of 8*11 paper and lay them on top of one another so that every point on the top paper corresponds with a point on the bottom paper. Now crumple the top piece of paper in anyway that you wish and place it back on top. B's theorem tells us that there must be a point which has not moved, i.e. which lies exactly above the same point that it did initially.

Last example. Consider a cupful of coffee. Each point is somewhere in 3-dimensional space. Stir. At least one point ends up in the same place as it began. [Note earlier I had molecule in place of point but the theorem requires continuity and molecules are finite so the wording was a bit misleading. Thanks to a number of readers who pointed out the error.]

Aside from these intriguing examples, B.'s theorem and Kakutani's extension are used extensively in economics such as to prove the existence of a Nash equilibrium.

Interesting piece of trivia from Kakutani's life. His father made him study literature and the arts before allowing him to take up mathematics. Today Kakutani's daughter, Michiko, is the chief book critic for The New York Times.

Here is an intuitive proof of the monk problem. Imagine that there are two monks, one going down and one going up, each beginning on the same day at sunrise. At some point in the day the hiker's must meet! QED.