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Must one of my favorite geometrical theorems come to be regarded as "trivial" or "obvious", when it is shown to have a really short and easy proof? Let C denote any closed unbounded subset of the Euclidean plane, whose interior is non-empty. One can visualize the immense variety of point sets that qualify as examples of C, and they all have positive or infinite 2-dimensional Lebesgue measure. But we have the theorem: "If C is convex, then C has infinite 2-dimensional Lebesgue measure".-Proof: Let p be an interior point of C. Then there exists a subset of C which is a closed disk D, centered at p, whose diameter is positive. Let q be a point of C not belonging to D. Consider the isoceles triangle T whose base is a diameter of D and whose opposite vertex is q. The staight line segment pq is a median and an altitude of T. T is a subset of C and the unboundedness of C implies that q can be so chosen as to make the area of T arbitrarily large-Q.E.D. Note that both the theorem and its short proof generalize readily to higher dimensional Euclidean spaces. I first realized that there was such a theorem after managing to come up with a long and unwieldy proof of it.

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closed as not a real question by Todd Trimble, Noah Schweber, Anthony Quas, Lee Mosher, Felipe Voloch May 26 '13 at 16:16

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

I don't think this question is appropriate for MO, but: in defense of triviality, I'd point to the fact that part of the continuing goal of mathematics is to develop the right intuitions for its results -- so results should become easier over time. In some sense, I think the hope tends to be that theorems will become "trivial" in some sense, because that means mathematical intuition will have evolved to the point that those theorems will now seem inescapably true. – Noah Schweber May 26 '13 at 15:26
Many if not all such elementary proofs may look obvious once discovered. But so what? One can still take pleasure in them. This question seems more suitable for a bull session over beers than a good question for MO. Voted to close. – Todd Trimble May 26 '13 at 15:27
What IS the question?! – Amir Asghari May 26 '13 at 15:55
The only question I saw was the first sentence, which ends in a question mark. Viz., must we regard one of Garabed's favorite geometrical theorems as trivial? – Todd Trimble May 26 '13 at 16:02

1 Answer 1

The words "obvious" and "trivial" relate more to the prevailing mathematical culture than to the statement itself. Yes, the theorem you quoted is obvious in the current culture, and so is Euclid's proof, but not because the proofs are short and simple, rather because in both cases this particular way of argument became totally standard and repeats in so many places that it is essentially the first thing that comes to your mind when you look at the problem. Of course, I'm talking here of "an average mathematician". Each particular individual can manage to stay away from some aspects of general mathematical culture for a long while and then completely routine things can come as a big surprise to him (I experienced a few such surprises myself and, I believe, this experience is shared by many people). There is nothing wrong with stopping on the road and admiring the wheels of the passing cars for a while. As a matter of fact, this is quite a good exercise for your brain, IMHO (admiring=thinking of how it works and how else you could make it, not just staring). Still, most people won't share your excitement even when you show them some of the latest Lockheed Martin products instead of the car wheel: everybody got so used to the idea of the flying machines that no one is particularly impressed by anything. The ability to perceive things you see every day as unusual (and most of them are highly unusual) has to be maintained consciously by routine exercising more often than not. The same is true for the world of mathematics.

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Thanks for your responses. I'm not surprised my question got closed out but I hoped that I could get it through as a "soft question". – Garabed Gulbenkian May 27 '13 at 15:28

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