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Generalized versions of Ilmanen lemma: Insertion of $ C^{1,\omega} $ or $ C^{1,\omega}_{{\rm loc}} $ functions
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 …