Set of examples
정리나 성질의 확장, 일반화
- Heron optical maxmin - steiner network
- Integral의 정의/개념,
비슷한 증명 Idea
- Heron's maxmin : Torricelli-Fermat-Steiner's Point, Fagnano-Schwarts triangle,
그러나 general case 가 특별한 경우와 수학적 본질이 많이 다른 경우가 있다. 어떤 경우는 특이한 경우가 더 먼저 증명되고 일반화된 정리가 증명되고 어떤 경우는 그 반대다.
- (반대 경우)
The original phrasing was as follows:
Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?
Poincaré never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the Poincaré conjecture. Here is the standard form of the conjecture:
Every simply connected compact 3-manifold (without boundary) is homeomorphic to a 3-sphere.
The conjecture can be generalized to other dimensions, and was solved much earlier (it has a substantially different flavor)
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