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The length of any curve that goes from A to B (other than the line segment AB) is greater than the length of that segment. This statement may be a theorem? Or, necessarily, is an axiom? It seems to me that any attempt to build a theorem eventually turning into begging.

Thank you very much!

Paulo Argolo

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The proof of Oprea's "Theorem 1.1.11" here is instructive. –  J. M. Nov 24 '10 at 3:01

2 Answers 2

It's a consequence of the so-called triangle inequality.

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This is a (simple) theorem. See http://en.wikipedia.org/wiki/Arc_length#Definition.

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@Ricky: Not using the definition of curve length the fact that the lower curve from A to B is the line segment AB? –  Paulo Argolo Nov 23 '10 at 23:55
    
What do you mean by "the lower curve from A to B"? And what about that fact? –  Ricky Demer Nov 23 '10 at 23:58
    
@Ricky: I'm trying to say this: to establish the definition of length of a curve is allowed, a priori, that the curve of shortest length, which goes from A to B, is exactly the line segment AB. –  Paulo Argolo Nov 24 '10 at 0:16
    
No, you can have a length of a curve in a non-Eucledian metric space so that the line segment either won't be the shortest curve, or won't be the only shortest curve. –  trutheality Nov 24 '10 at 2:45

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