How do control the boundary regularity of the Legendre transformation domain from a convex function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:15:23Z http://mathoverflow.net/feeds/question/66748 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66748/how-do-control-the-boundary-regularity-of-the-legendre-transformation-domain-fro How do control the boundary regularity of the Legendre transformation domain from a convex function fible 2011-06-02T15:52:51Z 2011-06-02T15:58:59Z <p>Let f(x) be a strongly convex smooth function (its Hessian matrix is positive definite) defined in a convex domain D, introduce the Legendre transformation $$x=(x_1,...,x_n)\rightarrow (\xi_1,...,\xi_n),\xi_i=\frac{\partial f}{\partial x_i},$$ $$u(\xi_1,...,\xi_n)=x_i\xi_i-f$$ The Legendre transformation domain W is defined by: $$W=((\xi_1,...,\xi_n)|\xi_i=\frac{\partial f}{\partial x_i}, x\in D )$$ I want to know the regularity of the boundary of W, (can assume the domain W is bounded) what conditions to make the boundary $\partial W$ smooth or $C^2$?</p>