zbMATH — the first resource for mathematics

Global geometry and $$C^1$$ convex extensions of 1-jets. (English) Zbl 1403.26009
Summary: Let $$E$$ be an arbitrary subset of $$\mathbb{R}^n$$ (not necessarily bounded) and $$f:E\rightarrow\mathbb{R},\,G:E\rightarrow\mathbb{R}^n$$ be functions. We provide necessary and sufficient conditions for the 1-jet $$(f,G)$$ to have an extension $$(F,\nabla F)$$ with $$F:\mathbb{R}^n\rightarrow\mathbb{R}$$ convex and $$C^{1}$$. Additionally, if $$G$$ is bounded we can take $$F$$ so that $$\operatorname{Lip}(F)\lesssim\|G\|_{\infty}$$. As an application we also solve a similar problem about finding convex hypersurfaces of class $$C^1$$ with prescribed normals at the points of an arbitrary subset of $$\mathbb{R}^n$$.

MSC:
 26B05 Continuity and differentiation questions 26B25 Convexity of real functions of several variables, generalizations 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces)
Full Text:
References:
 [1] 10.1112/plms/pds099 · Zbl 1282.41008 [2] 10.1112/plms.12006 · Zbl 1379.54014 [3] 10.1016/j.jmaa.2006.03.088 · Zbl 1116.49011 [4] 10.1016/j.jfa.2017.12.007 · Zbl 1393.58006 [5] 10.1090/S0002-9947-01-02756-8 · Zbl 0973.46025 [6] 10.1090/proc/14012 · Zbl 1406.54009 [7] 10.4007/annals.2005.161.509 · Zbl 1102.58005 [8] 10.4007/annals.2006.164.313 · Zbl 1109.58016 [9] 10.1090/S0273-0979-08-01240-8 · Zbl 1207.58011 [10] 10.1007/s00039-016-0366-7 · Zbl 1353.58004 [11] 10.4171/RMI/938 · Zbl 1366.65010 [12] 10.4310/jdg/1090348111 · Zbl 1068.53009 [13] 10.1007/BF02790231 · Zbl 0091.28103 [14] 10.24033/asens.1361 · Zbl 0415.31001 [15] 10.2307/2001721 · Zbl 0712.49010 [16] 10.1007/s00039-009-0027-1 · Zbl 1196.54035 [17] ; Rockafellar, Convex analysis. Princeton Mathematical Series, 28, (1970) [18] 10.1080/02331937908842605 · Zbl 0439.26007 [19] 10.2307/1989708 · JFM 60.0217.01 [20] ; Yan, J. Convex Anal., 21, 965, (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.