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I'm having trouble with your "therefore" conclusion. The projection of the perimeter of the inner box could easily be shorter than the sum of the lengths of the projections of each edge. It seems like you need a less obvious (to me, anyway) statement about the projection of a collection of lines.

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By projected perimeter I mean the sum of lengths of projections of all edges. I'm not canceling them out or anything. Think of the box as a graph, its projection is another graph that happens to lie on a straight line, but we can measure the sum of its edges regardless.

It's not completely obvious why the projected perimeter of the inner box is bounded by the projected perimeter of the outer box, but it's a statement about one-dimensional segments that's easy enough to check.

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