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I would like to have an inverse (or/and) implicite function theorem for DC-functions. It seems that I have right definitions, but I fail to prove it...

Definitions:

  • Let $h:\mathbb R^n\to\mathbb R$ be a convex function and $x\in\mathbb R^n$, the set of all linear functionals $\ell:\mathbb R^n\to\mathbb R$ such that $$h(y)\ge h(x)+\ell(y-x)$$ is called subdifferential of $h$ at $x$ --- it will be denoted as $\partial_{x}h$. (In general $\partial_{x}h$ is a nonempty bounded convex set)

  • $f:\mathbb R^n\to\mathbb R$ is called DC-function if it is a difference between two convex functions.

  • $F:\mathbb R^n\to\mathbb R^k$ is called DC-map each coordinate function of $F$ is DC.

  • $x\in\mathbb R^n$ is called regular value of a DC-function $f:\mathbb R^n\to\mathbb R$ if of $f=h_1-h_2$ for some convex functions $h_1$ and $h_2$ and $\partial_x h_1 + (-\partial_x h_2)\not\ni0$. Here $+$ denotes Minkowski sum and $(-\partial_x h_2)$ is reflection of $\partial_x h_2$ in the origin.

  • $x\in\mathbb R^n$ is called regular value of a DC-map $f:\mathbb R^n\to\mathbb R^k$ if $x$ is a regular value of $\ell\circ F$ for any non-zero linear map $\ell:\mathbb R^k\to\mathbb R$.

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According to this paper an implicit function theorem for delta-convex functions is given in Theorem 4.4 of L. Veselý and L. Zajíček, "Delta-convex mappings between Banach spaces and applications." Dissertationes Math. (Rozprawy Mat.) 289 (1989).

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