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 Jun 5 comment How do control the boundary regularity of the Legendre transformation domain from a convex function 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. Jun 3 awarded Teacher Jun 3 revised surfaces of constant centro-affine curvature corrected spelling Jun 3 revised surfaces of constant centro-affine curvature corrected spelling Jun 3 answered surfaces of constant centro-affine curvature Jun 2 awarded Editor Jun 2 revised How do control the boundary regularity of the Legendre transformation domain from a convex function improved formatting Jun 2 asked How do control the boundary regularity of the Legendre transformation domain from a convex function May 30 answered Solutions to a Monge-Ampère equation on the simplex