Functions on a convex set which are both $ ω$-semiconvex and $ ω$-semiconcave II
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909146313392128 |
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| author | Kryštof, Václav |
| author_facet | Kryštof, Václav |
| contents | In a recent article (2022) we proved with L. Zajíček that if $ G\subset\R^n $ is an unbounded open convex set that does not contain a translation of a convex cone with non-empty interior, then there exist $ f:G\to\R $ and a concave modulus $ ω$ such that $ \lim_{t\to\infty}ω(t)=\infty $, $ f $ is both semiconvex and semiconcave with modulus $ ω$ and $ f\notin C^{1,ω}(G) $. Here we improve the previous result as follows: If $ G $ is as above and $ ω(t)=t^α $ for some $ α\in(0,1) $, then there exists $ f:G\to\R $ that is both semiconvex and semiconcave with modulus $ ω$ and $ f\notin C^{1,α}(G) $. This result has immediate consequences concerning a first-order quantitative converse Taylor theorem and the problem whether $ f\in C^{1,α}(G) $ whenever $ f $ is smooth in a corresponding sense on all lines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14901 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Functions on a convex set which are both $ ω$-semiconvex and $ ω$-semiconcave II Kryštof, Václav Classical Analysis and ODEs In a recent article (2022) we proved with L. Zajíček that if $ G\subset\R^n $ is an unbounded open convex set that does not contain a translation of a convex cone with non-empty interior, then there exist $ f:G\to\R $ and a concave modulus $ ω$ such that $ \lim_{t\to\infty}ω(t)=\infty $, $ f $ is both semiconvex and semiconcave with modulus $ ω$ and $ f\notin C^{1,ω}(G) $. Here we improve the previous result as follows: If $ G $ is as above and $ ω(t)=t^α $ for some $ α\in(0,1) $, then there exists $ f:G\to\R $ that is both semiconvex and semiconcave with modulus $ ω$ and $ f\notin C^{1,α}(G) $. This result has immediate consequences concerning a first-order quantitative converse Taylor theorem and the problem whether $ f\in C^{1,α}(G) $ whenever $ f $ is smooth in a corresponding sense on all lines. |
| title | Functions on a convex set which are both $ ω$-semiconvex and $ ω$-semiconcave II |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2403.14901 |