What are examples of theorems get extensions based on simple observation? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T12:22:15Z http://mathoverflow.net/feeds/question/10459 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10459/what-are-examples-of-theorems-get-extensions-based-on-simple-observation What are examples of theorems get extensions based on simple observation? lemega 2010-01-02T04:21:05Z 2012-06-07T10:19:53Z <p>Here are some examples illustrate what I meant:</p> <p>Bonnet-Myers:Bonnet in 1855 proved n=2 case, Myers in 1941 extended to any dimension using the same idea.</p> <p>Bishop-Gromov Volume comparison: Bishop knew the result inside cut locus in 1963, Gromov made a simple observation that beyond cut locus it is still true.</p> <p>Ok, you got what I mean. </p> http://mathoverflow.net/questions/10459/what-are-examples-of-theorems-get-extensions-based-on-simple-observation/10510#10510 Answer by Joseph Malkevitch for What are examples of theorems get extensions based on simple observation? Joseph Malkevitch 2010-01-02T19:10:00Z 2010-01-02T19:10:00Z <p>Perhaps Euler's polyhedral formula:</p> <p>V(vertices) + F(faces) - E(edges) = 2</p> <p>provides an example of what you mean?</p> <p>Euler did not give a proper proof but shortly thereafter this result inspired huge advances that had dramatic effects on the evolution of geometry, topology, convexity, and what today is called graph theory. </p> <p>One measure of how rich this topic is can be seen from the many types and styles of proofs that can be found for this result collected below by David Eppstein:</p> <p><a href="http://www.ics.uci.edu/~eppstein/junkyard/euler/" rel="nofollow">http://www.ics.uci.edu/~eppstein/junkyard/euler/</a></p>