complex convex functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T18:26:33Z http://mathoverflow.net/feeds/question/91738 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91738/complex-convex-functions complex convex functions Zhipeng Lu 2012-03-20T17:08:02Z 2012-03-20T17:41:09Z <p>I call a function f defined and valued on a domain A in the plane convex if it maps convex areas to convex areas. Some obvious example of convex functions. If f is also a bijection, what can we say more about it? I guessed that if f is a diffeomorphism(C2) of the complex plane then it is linear, say az+b. This is related to a quadratic form of its differentials fxx,fyy,fxy. However, I cannot strictly confirm this. Is there any classical analysis addressing with this question? If any, please help show the sources.</p> http://mathoverflow.net/questions/91738/complex-convex-functions/91740#91740 Answer by alvarezpaiva for complex convex functions alvarezpaiva 2012-03-20T17:33:38Z 2012-03-20T17:33:38Z <p>I think that <em>A bijection from the plane to itself that sends convex sets into convex sets is an affine transformation.</em></p> <p>Indeed, it must not be hard to see that such bijection sends lines to lines (into what would it send a half-plane given that it is convex and has a convex complement??) and then one applies the fundamental theorem of affine geometry. </p>