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Consider the transform of nonnegative continuous concave positive homogenuous of first order function $f(x)$, $x \in \mathbb R^n_+$, $f \not\equiv 0$ given by $$ f^\times(y)= \inf \left\{ \left. \frac{\langle x,y \rangle}{f(x)} \, \right|\, x \colon f(x) > 0 \right\}, $$ where $\langle x, y \rangle = x_1y_1+\ldots+x_n y_n$. It can easily be shown that $f^\times$ is concave, positive-homogenuous of first order, nonnegative,$f^\times \not\equiv 0$. It is also continuous, but the proof of continuity isn't trivial. Furthermore $f^{\times \times} = f$ and $f(x)f^\times(y) \leqslant \langle x,y \rangle $. In few words, this is an analogue of Fenchel transorm (if I'm not mistaken it was introduced by G.Kleiner).

But is it true that if $f \in C^k(\mathrm{int}\, \mathbb R^n_+)$ then $f^\times\in C^k(\mathrm{int}\, \mathbb R^n_+)$? Looking at the proof of continuity of $f^\times$ I think that this question is very nontrivial. Any ideas how to prove this are appreciated.

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The answer is negative. Even if $f$ is a real analytic function on $\mathbb R^n_+$ the function $f^\times$ need not to be $C^1(\mathop{\mathrm{int}}\mathbb R^n_+)$. A simple counter-example is $f(x) = x_1+x_2$. Then $f^\times(y) = \min(y_1,y_2)$.

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