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Keywords:
Ilmanen lemma; $C^{1,\omega}$ function; semiconvex function with general modulus
Summary:
We prove that for a normed linear space $X$, if $f_1\colon X\to\mathbb{R}$ is continuous and semiconvex with modulus $\omega$, $f_2\colon X\to\mathbb{R}$ is continuous and semiconcave with modulus $\omega$ and $f_1\leq f_2$, then there exists $f\in C^{1,\omega}(X)$ such that $f_1\leq f\leq f_2$. Using this result we prove a generalization of Ilmanen lemma (which deals with the case $\omega(t)=t$) to the case of an arbitrary nontrivial modulus $\omega$. This generalization (where a $C^{1,\omega}_{{loc}}$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.
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