How do control the boundary regularity of the Legendre transformation domain from a convex function - MathOverflow most recent 30 from http://mathoverflow.net2013-05-19T21:15:23Zhttp://mathoverflow.net/feeds/question/66748http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/66748/how-do-control-the-boundary-regularity-of-the-legendre-transformation-domain-froHow do control the boundary regularity of the Legendre transformation domain from a convex functionfible2011-06-02T15:52:51Z2011-06-02T15:58:59Z
<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>