Abstract
We prove that if \(u:\mathbb{R}^{n}\to \mathbb{R}\) is strongly convex, then for every \(\varepsilon >0\) there is a strongly convex function \(v\in C^{2}(\mathbb{R}^{n})\) such that \(|\{u\neq v\}|<\varepsilon \) and \(\Vert u-v\Vert _{\infty}<\varepsilon \).
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Funding
D.A. was supported by grant PID2022-138758NB-I00. M.D. was supported by NSF Award No. 2103209. P.H. was supported by NSF grant DMS-2055171 and by Simons Foundation grant 917582.
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Azagra, D., Drake, M. & Hajłasz, P. \(\mathbf{C^{2}}\)-Lusin approximation of strongly convex functions. Invent. math. (2024). https://doi.org/10.1007/s00222-024-01252-6
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DOI: https://doi.org/10.1007/s00222-024-01252-6