Can't put a big box in a small box

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Lopsidation
Posts: 183
Joined: Tue Oct 27, 2009 11:29 pm UTC

Can't put a big box in a small box

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.

Qaanol
The Cheshirest Catamount
Posts: 3069
Joined: Sat May 09, 2009 11:55 pm UTC

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

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

Spoiler:
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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