Can't put a big box in a small box

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Can't put a big box in a small box

Postby Lopsidation » Thu Apr 16, 2015 3:51 pm UTC

1. Rectangle A contains rectangle B. Must the perimeter of A >= the perimeter of B?

2. The 3-dimensional box (ie rectangular prism) A contains the 3-dimensional box B. Must the surface area of A be at least that of B? What about the sum of edge lengths -- can B be more edgy than A?

3. The n-dimensional box A contains the n-dimensional box B. The k-dimensional area of a box is the sum of the areas of all k-dimensional faces. For all n and k, must A's k-dimensional area be at least as large as B's?

I know the answer to the first two problems, but not the last one.

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Re: Can't put a big box in a small box

Postby Qaanol » Fri Apr 17, 2015 5:23 am UTC

Here’s a quick, probably-could-be-made-rigorous, approach to the the k = n-1 subproblem:

Pick a point inside the inner box and project outward from it. In other words, map each point on the surface of the inner box to the point on the surface of the outer box colinear with it and the chosen point. This cannot shrink the surface, therefore the (n-1)-dimensional area of the outer box is at least as large as that of the inner box.
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