Convex Polygons - Page 5

davar55 · 2007-02-22, 18:56

My pleasure.

I was hoping someone would suggest that this proposition
could be generalized. I can think of at least two possible ways:

Any convex polyhedron completely containing (but not
identical to) another convex polyhedron necessarily has
a greater surface area.

Any closed convex planar region completely containing (but not
identical to) another closed convex planar region necessarily
has a greater perimeter.

The first I think can be proved by an extension of the polygon
proof. The second seems to need a limit proof or some
topological argument. I leave both as open questions.

davieddy · 2007-02-25, 14:34

The answer seems obvious, and when you read the solution
you understand why:)))

David

2007-02-25, 14:34	#46
davieddy "Lucan" Dec 2006 England 2·3·13·83 Posts	I Like This Problem The answer seems obvious, and when you read the solution you understand why:))) David

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2007-02-22, 18:56	#45
davar55 May 2004 New York City 1000010001011₂ Posts	My pleasure. I was hoping someone would suggest that this proposition could be generalized. I can think of at least two possible ways: Any convex polyhedron completely containing (but not identical to) another convex polyhedron necessarily has a greater surface area. Any closed convex planar region completely containing (but not identical to) another closed convex planar region necessarily has a greater perimeter. The first I think can be proved by an extension of the polygon proof. The second seems to need a limit proof or some topological argument. I leave both as open questions.