The Ham Sandwich Theorem

http://en.wikipedia.org/wiki/Ham_sandwich_theorem

by jbert • 17 years ago

6 4 17 years ago

en.wikipedia.org

Having seen the Hairy Ball theorem posted here a few hours ago I was wondering how long it would be before someone posted the Ham Sandwich Theorem.

The interesting thing is that it works in all dimensions. In 8 dimensional space, for example, 8 convex (or even non-convex, and even non-connected!) bodies can simultaneously be cut in half by a 7 dimensional hyperplane. I actually do that for my work - it's not purely theoretical.

TrevorJ• 17 years ago

Is it possible to shed some light on the practical application you are using?

bendotc• 17 years ago

Eight dimensional hypersandwich making, obviously. No matter how you want the eight ingredients oriented, that sandwich will be cut in two nice, even halves.

RiderOfGiraffes2• 17 years ago

Um, no. Sorry, commercial in confidence.

However, let me give an analogy. Aeroplanes live in X-dimensional phase space. Three space coordinates, three attitude coordinates (pitch/roll/yaw), three velocity coordinates (usually, but not always, aligned with the attitude) at a given time. Thus we need to deal with a map from one dimension - time - to nine dimensions. If we simplify some things, then look also at the other rates of change, we get a complex representation of non-independent variables (velocity is just rateof change of position, for example), but in the complexity of the representation, some of the behavior becomes simplified.

Now take clusters of these measurements and try to identify from among them which ones all correspond to the same object. Another analogy - not what I do - is to take lots of position measurements widely spaced in time of planes in an acrobatic airshow, and try to identify the trajectories of individual 'planes.

Now do it all in a noisy environment with instrument error, and if you've arranged it just right, finding equal sized cuts of convex or near convex bodies can correspond to separating the plots of different objects.

Actually, I can get a function that lifts, supports and separates convex bodies. I called it the "Playtex Function", but because of the work I do I couldn't publish.

All that is obscure, I know, but it gives some sense of what I do. Sorry I can't be clearer.

The Ham Sandwich Theorem

Related Stories