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 $C^2$?
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$\begingroup$ If $f$ is strongly convex and defined in a nbd of $\bar D$, it's gradient $\nabla f$ is a (monotone) diffeomorphism, so $\partial W$ is $C^2$ if $\partial D$ is $C^2$ and $f$ is $C^3$ on a nbd of $\bar D$. $\endgroup$– Pietro MajerCommented Jun 2, 2011 at 17:26
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$\begingroup$ In my question, f(x) may be defined on the whole $\mathbb{R}^n$ (this situation is my interest), for example, the following function (known as hyperbolic affine hypersphere): $$f(x_1,...,x_n)=\frac{1}{x_1\cdots x_n}, x_i>0, 1\leq i\leq n.$$ Choose suitable coordinates, this graph can be represented by another function $\tilde{f}$ defined on the whole $\mathbb{R}^n$, and the Legendre transformation domain of $\tilde{f}$ is a simplex. $\endgroup$– fibleCommented Jun 5, 2011 at 2:28
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