Abstract

Let XX denote Rn\mathbb{R}^n or, more generally, a Hilbert space. Given an arbitrary subset CC of XX and a collection H\mathcal{H} of affine hyperplanes of XX such that every H∈HH\in\mathcal{H} passes through some point xH∈Cx_{H}\in C, and C={xH:H∈H}C=\{x_H: H\in\mathcal{H}\}, what conditions are necessary and sufficient for the existence of a C1,1C^{1,1} convex hypersurface SS in XX such that HH is tangent to SS at xHx_H for every H∈HH\in\mathcal{H}? In this paper we give an answer to this question. We also provide solutions to similar problems for convex hypersurfaces of class C1,ωC^{1, \omega} in Hilbert spaces, and for convex hypersurfaces of class C1,αC^{1, \alpha} in superreflexive Banach spaces having equivalent norms with moduli of smoothness of power type 1+α1+\alpha, α∈(0,1]\alpha\in (0, 1].

Author information

Contact details are reproduced from the original publication and may be historical.

Daniel Azagra

ICMAT (CSIC-UAM-UC3-UCM), Madrid, Spain
and: Dep. de Análisis Matemático y Matemática Aplicada, Facultad Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain

azagra@mat.ucm.es

Keywords

  • Convex body
  • convex function
  • Whitney extension theorem
  • differentiability
  • signed distance function

Mathematics Subject Classification

6B05, 26B25, 52A05, 52A20

References

  1. D. Azagra, E. Le Gruyer, C. Mudarra: Explicit formulas for C 1,1 and Cconv 1,ω extensions of 1-jets in Hilbert and superreflexive spaces, J. Funct. Anal. 274 (2018) 3003–3032.https://doi.org/10.1016/j.jfa.2017.12.007Google Scholar
  2. D. Azagra, C. Mudarra: Whitney Extension Theorems for convex functions of the classes C 1 and C 1,ω, Proc. London Math. Soc. 114(1) (2017) 133–158.https://doi.org/10.1112/plms.12006Google Scholar
  3. D. Azagra, C. Mudarra: An Extension Theorem for convex functions of class C 1,1 on Hilbert spaces, J. Math. Anal. Appl. 446 (2017) 1167–1182.https://doi.org/10.1016/j.jmaa.2016.09.015Google Scholar
  4. D. Azagra, C. Mudarra: Global geometry and C 1 convex extensions of 1-jets, Analysis and PDE 12(4) (2019) 1065–1099.https://doi.org/10.2140/apde.2019.12.1065Google Scholar
  5. H. Bauschke, P. L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd ed., Springer, New York (2017).https://doi.org/10.1007/978-3-319-48311-5Google Scholar
  6. E. Bierstone, P. Milman, W. Pawlucki: Differentiable functions defined on closed sets. A problem of Whitney, Inventiones Math. 151(2) (2003) 329–352.https://doi.org/10.1007/s00222-002-0255-6Google Scholar
  7. J. M. Borwein, S. Fitzpatrick: Existence of nearest points in Banach spaces, Canad. J. Math. 41(4) (1989) 702–720.https://doi.org/10.4153/CJM-1989-032-7Google Scholar
  8. A. Brudnyi, Y. Brudnyi: Methods of Geometric Analysis in Extension and Trace Problems. Volumes 1 and 2., Monographs in Mathematics 102 and 103, Birkhäuser, Basel (2012).https://doi.org/10.1007/978-3-0348-0209-3Google Scholar
  9. Y. Brudnyi, P. Shvartsman: Whitney’s extension problem for multivariate C 1,ω - functions, Trans. Amer. Math. Soc. 353 (2001) 2487–2512.https://doi.org/10.1090/S0002-9947-01-02756-8Google Scholar
  10. F. H. Clarke, R. J. Stern, P. R. Wolenski: Proximal smoothness and the lower-C 2 property, J. Convex Analysis 2(1-2) (1995) 117–144.https://doi.org/10.68381/jca02008Google Scholar
  11. G. Colombo, L. Thibault: Prox-regular sets and applications, in: Handbook of Nonconvex Analysis and Applications, D. Gao and D. Motreanu (eds.), International Press, Somerville (2010) 99–182.Google Scholar
  12. A. Daniilidis, M. Haddou, E. Le Gruyer, O. Ley: Explicit formulas for C 1,1 Glaeser-Whitney extensions of 1-fields in Hilbert spaces, Proc. Amer. Math. Soc. 146 (2018) 4487–4495.https://doi.org/10.1090/proc/14012Google Scholar
  13. M. C. Delfour, J. P. Zolésio: Shape analysis via oriented distance functions, J. Funct. Anal. 123 (1994) 129–201.https://doi.org/10.1006/jfan.1994.1086Google Scholar
  14. R. Deville, G. Godefroy, V. Zizler: Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific & Technical, Harlow; copublished with John Wiley & Sons, New York (1993).Google Scholar
  15. M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler: Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics, Springer, New York (2011).https://doi.org/10.1007/978-1-4419-7515-7Google Scholar
  16. C. Fefferman: A sharp form of Whitney’s extension theorem, Ann. of Math. (2) 161(1) (2005) 509–577.https://doi.org/10.4007/annals.2005.161.509Google Scholar
  17. C. Fefferman: Whitney’s extension problem for C m, Ann. of Math. (2)(1) 164 (2006) 313–359.https://doi.org/10.4007/annals.2006.164.313Google Scholar
  18. C. Fefferman: Whitney’s extension problems and interpolation of data, Bull. Amer. Math. Soc. (N.S.) 46(2) (2009) 207–220.https://doi.org/10.1090/S0273-0979-08-01240-8Google Scholar
  19. C. Fefferman, A. Israel, G. K. Luli: Sobolev extension by linear operators, J. Amer. Math. Soc. 27(1) (2014) 69–145.https://doi.org/10.1090/S0894-0347-2013-00763-8Google Scholar
  20. C. Fefferman, A. Israel, G. K. Luli: Finiteness principles for smooth selection, Geom. Funct. Anal. 26(2) (2016) 422–477.https://doi.org/10.1007/s00039-016-0366-7Google Scholar
  21. C. Fefferman, A. Israel, G. K. Luli: Interpolation of data by smooth nonnegative functions, Rev. Mat. Iberoam. 33(1) (2017) 305–324.https://doi.org/10.4171/rmi/938Google Scholar
  22. C. Fefferman, B. Klartag: Fitting a C m -smooth function to data I, Ann. of Math. 169(1) (2009) 315–346.https://doi.org/10.4007/annals.2009.169.315Google Scholar
  23. S. Fitzpatrick: Differentiation of real-valued functions and continuity of metric projections, Proc. Amer. Math. Soc. 91(4) (1984) 544–548.https://doi.org/10.1090/S0002-9939-1984-0746087-9Google Scholar
  24. S. Fitzpatrick, R. R. Phelps: Differentiability of the metric projection in Hilbert space, Trans. Amer. Math. Soc. 270 (1982) 483–501.https://doi.org/10.1090/S0002-9947-1982-0645326-5Google Scholar
  25. M. Ghomi: Strictly convex submanifolds and hypersurfaces of positive curvature, J. Differential Geom. 57 (2001) 239–271.https://doi.org/10.4310/jdg/1090348111Google Scholar
  26. M. Ghomi: Optimal smoothing for convex polytopes, Bull. London Math. Soc. 36 (2004) 483–492.https://doi.org/10.1112/S0024609303003059Google Scholar
  27. M. Ghomi, R. Howard: Tangent cones and regularity of real hypersurfaces, J. Reine Angew. Math. 697 (2014) 221–247.https://doi.org/10.1515/crelle-2013-0091Google Scholar
  28. G. Glaeser: Etudes de quelques algèbres tayloriennes, J. d’Analyse 6 (1958) 1–124.https://doi.org/10.1007/BF02790231Google Scholar
  29. J.-B. Hiriart-Urruty: Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res. 4 (1979) 79–97.https://doi.org/10.1287/moor.4.1.79Google Scholar
  30. J.-B. Hiriart-Urruty: New concepts in non differentiable programming, Bull. Soc. Math. France Mém. 60 (1979) 57–85.https://doi.org/10.24033/msmf.261Google Scholar
  31. R. B. Holmes: Smoothness of certain metric projections on Hilbert space, Trans. Amer. Math. Soc. 183 (1973) 87–100.https://doi.org/10.1090/S0002-9947-1973-0326252-2Google Scholar
  32. B. Klartag, N. Zobin: C 1 extensions of functions and stabilization of Glaeser refinements, Rev. Mat. Iberoam. 23(2) (2007) 635–669.https://doi.org/10.4171/rmi/507Google Scholar
  33. S. G. Krantz, H. R. Parks: Distances to C k hypersurfaces, J. Diff. Equations 40 (1981) 116–120.https://doi.org/10.1016/0022-0396(81)90013-9Google Scholar
  34. E. Le Gruyer: Minimal Lipschitz extensions to differentiable functions defined on a Hilbert space, Geom. Funct. Anal 19(4) (2009) 1101–1118.https://doi.org/10.1007/s00039-009-0027-1Google Scholar
  35. K. R. Lucas: Submanifolds of dimension n − 1 in E n with normals satisfying a Lipschitz condition, in: Studies in Eigenvalue Problems, Technical Report 18, Kansas University, Dept. of Mathematics (1957), http://hdl.handle.net/2027/mdp.39015017419931.Google Scholar
  36. G. Pisier: Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975) 236–350.https://doi.org/10.1007/BF02760337Google Scholar
  37. R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352(11) (2000) 5231–5249.https://doi.org/10.1090/S0002-9947-00-02550-2Google Scholar
  38. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  39. P. Shvartsman: On Sobolev extension domains in Rn, J. Funct. Anal. 258(7) (2010) 2205–2245.https://doi.org/10.1016/j.jfa.2010.01.002Google Scholar
  40. P. Shvartsman: Sobolev L2p -functions on closed subsets of R2, Adv. Math. 252 (2014) 22–113.https://doi.org/10.1016/j.aim.2013.10.017Google Scholar
  41. P. Shvartsman: Whitney-type extension theorems for jets generated by Sobolev functions, Adv. Math. 313 (2017) 379–469.https://doi.org/10.1016/j.aim.2017.04.009Google Scholar
  42. P. Shvartsman, N. Zobin: On planar Sobolev Lm p -extension domains, Adv. Math. 287 (2016) 237–346.https://doi.org/10.1016/j.aim.2015.08.031Google Scholar
  43. J. C. Wells: Differentiable functions on Banach spaces with Lipschitz derivatives, J. Differential Geometry 8 (1973) 135–152.https://doi.org/10.4310/jdg/1214431488Google Scholar
  44. H. Whitney: Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934) 63–89.https://doi.org/10.1090/S0002-9947-1934-1501735-3Google Scholar
  45. M. Yan: Extension of convex function, J. Convex Analysis 21(4) (2014) 965–987.https://doi.org/10.68381/jca21052Google Scholar
  46. N. Zobin: Whitney’s problem on extendability of functions and an intrinsic metric, Adv. Math. 133(1) (1998) 96–132.https://doi.org/10.1006/aima.1997.1685Google Scholar

Suggested citation

D. Azagra, C. Mudarra. “Prescribing Tangent Hyperplanes to C^{1,1} and C^{1, ω} Convex Hypersurfaces in Hilbert and Superreflexive Banach Spaces.” Journal of Convex Analysis 27 (2020), No. 1, 79–102. https://doi.org/10.68381/jca27006

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press.