Does convex continuous mapping have a unique fixed point? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:15:57Z http://mathoverflow.net/feeds/question/72281 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72281/does-convex-continuous-mapping-have-a-unique-fixed-point Does convex continuous mapping have a unique fixed point? Danu 2011-08-07T12:00:15Z 2011-08-07T12:23:14Z <p>(Note: I have a very little knowledge in the area related to this question. (My research is more related to Combinatorics.) So, I apologize in advance if I say something wrong or trivial.) </p> <p>One version of the Brouwer fixed point theorem states that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point. Moreover, it is possible that many fixed points exist. </p> <p>One specific question I would like to ask is whether there exists a <em>unique</em> fixed point if we restrict the function to be <em>convex</em>. Any pointers to something related or counter examples would be appreciated. </p> <p>More generally, what are the restrictions on the functions that are sufficient to guarantee the uniqueness. The only result I'm aware of is Banach fixed point theorem which says that if the function is a contraction mapping (thus continuous) then there is a unique fixed point. Are there other restrictions like this? </p> http://mathoverflow.net/questions/72281/does-convex-continuous-mapping-have-a-unique-fixed-point/72283#72283 Answer by Jay Kangel for Does convex continuous mapping have a unique fixed point? Jay Kangel 2011-08-07T12:09:32Z 2011-08-07T12:09:32Z <p>As to the specific question in the next to last paragraph, the identity function is convex. Except for the zero dimensional case there are many fixed points.</p> http://mathoverflow.net/questions/72281/does-convex-continuous-mapping-have-a-unique-fixed-point/72284#72284 Answer by Gerald Edgar for Does convex continuous mapping have a unique fixed point? Gerald Edgar 2011-08-07T12:23:14Z 2011-08-07T12:23:14Z <p>Surely you can come up with a convex function $[0,1] \to [0,1]$, not the identity, with two fixed points.</p>