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 $$x=(x_1,...,x_n)\rightarrow (\xi_1,...,\xi_n),\xi_i=\frac{\partial f}{\partial x_i},$$ $$u(\xi_1,...,\xi_n)=x_i\xi_if$$ 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 $C^2$?
