24 May 08
The thread "Hard" is about a problem which, I think, demands a probabilistic solution but for which I can only find a sort of geometric one. This problem is the reverse: it's stated in geometric terms, and the only nice solution I can think of uses probability theory.
Consider an n-dimensional cube of volume 1. The intersection of this hypsercube and an (n-1)-dimensional hyperplane is an (n-1)-dimensional "polygon". What is the minimum possible volume of this polygon, and for what hyperplane is it attained?
(For example, consider the area of the polygon formed by the intersection of a plane and a cube, and generalise to more dimensions.)
24 May 08
2 edits
Originally posted by TheMaster37I forgot to add something important, namely that by "hyperplane" I mean "(vector) subspace of R^n". This means that you should choose coordinates for your cube with the origin inside. I'm happy if anyone does the case where the cube is centred at the origin (and thus the plane passes through its centre).
Hmm, the intersection of a plane and a cube can be a single point.
Generalising that to higher dimensions would mean the minimum area is 0.