

The Ham Sandwich Theorem - jbert
http://en.wikipedia.org/wiki/Ham_sandwich_theorem

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

~~~
RiderOfGiraffes
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.

