Abstract

We establish uniform estimates and properties of gauge functions for domains Ωε\Omega_\varepsilon, ε∈[0,1]\varepsilon\in[0,1], defined by the Minkowski sum Ωε=Ω0+εΩ\Omega_\varepsilon=\Omega_0+\varepsilon\Omega where Ω0\Omega_0 and Ω\Omega are convex and bounded subsets of Rn\mathbb{R}^n. These estimates are in fact needed when one deals with shape derivatives in PDE-constrained shape optimization problems using this Minkowski sum as a deformation as it is done in a recent paper of A. Boulkhemair and A. Chakib [On a shape derivative formula with respect to convex domains, J. Convex Analysis 21/1 (2014) 67–87] for example. We first show that this class of domains Ωε\Omega_\varepsilon satisfies the so-called uniform ball property which is equivalent to the positiveness of its reach. Then, we establish the said uniform estimates on the gauge function of Ωε\Omega_\varepsilon and its gradient as well as its hessian, with respect to the parameter ε\varepsilon.

Author information

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

Abdesslam Boulkhemair

Jean Leray Math. Lab., UFR-Sciences and Techniques, Nantes, France
and: UMR6629 CNRS, Nantes, France

boulkhemair-a@univ-nantes.fr

Abdelkrim Chakib

Applied Mathematics Team, Faculty of Sciences and Techniques, Sultan Moulay Slimane
University, Beni Mellal, Morocco

chakib.fstbm@gmail.com

Azeddine Sadik

(1) Department of Mathematics, Faculty of Sciences, Mohammed V University in Rabat, Rabat 10000, Morocco
(2) Jean Leray Math. Lab., UMR 6629 CNRS, UFR-Sciences and Techniques Nantes, France
(3) Applied Mathematics Team, Faculty of Sciences and Techniques, Sultan Moulay Slimane University, Beni Mellal, Morocco
(4) Besançon Mathematics Laboratory, University of Franche-Comté, UMR 6623 CNRS UBFC, Besançon, France

sadik.fstbm@gmail.com

Keywords

  • Convex domains
  • gauge functions
  • support functions
  • Minkowski sum
  • uniform ball condition
  • reach condition
  • uniform estimates

Mathematics Subject Classification

52A07; 52A30, 53A05, 49Q10

References

  1. I. Belegradek, Z. Jiang: Smoothness of Minkowski sum and generic rotations, J. Math. Analysis Appl. 450/2 (2017) 1229–1244.https://doi.org/10.1016/j.jmaa.2017.01.088Google Scholar
  2. A. Boulkhemair: On a shape derivative formula in the Brunn-Minkowski theory, SIAM J. Control Optimization 55/1 (2017) 156–171.https://doi.org/10.1137/15M1015844Google Scholar
  3. A. Boulkhemair, A. Chakib: On a shape derivative formula with respect to convex domains, J. Convex Analysis 21/1 (2014) 67–87.https://doi.org/10.68381/jca21004Google Scholar
  4. A. Boulkhemair, A. Chakib: Erratum: On a shape derivative formula with respect to convex domains, J. Convex Analysis 22 (2015) 901–903.https://doi.org/10.68381/jca22047Google Scholar
  5. A. Boulkhemair, A. Chakib, A. Nachaoui, A. A. Niftiyev, A. Sadik: On a numerical shape optimal design approach for a class of free boundary problems, Comp. Optim. Appl. 77 (2020) 509–537.https://doi.org/10.1007/s10589-020-00212-zGoogle Scholar
  6. A. Boulkhemair, A. Chakib, A. Sadik: Geometrical variations of a state-constrained functional on star-shaped domains, Advanced Math. Models Appl. 6/2 (2021) 73–88.Google Scholar
  7. A. Boulkhemair, A. Chakib, A. Sadik: On a shape derivative formula for star-shaped domains using Minkowski deformation, AIMS Mathematics 8/8 (2023) 19773–19793.https://doi.org/10.3934/math.20231008Google Scholar
  8. A. Boulkhemair, A. Chakib, A. Sadik: On numerical study of constrained coupled shape optimization problems based on a new shape derivative method, Numer. Methods Partial Diff. Equations 39 (2023) 2018–2059.https://doi.org/10.1002/num.22956Google Scholar
  9. A. Canino: On p-convex sets and geodesics, J. Diff. Equations 75/1 (1988) 118–157.https://doi.org/10.1016/0022-0396(88)90132-5Google Scholar
  10. P. Cannarsa, K. T. Nguyen: Exterior sphere condition and time optimal control for differential inclusions, SIAM J. Control Optimization 49/6 (2011) 2558–2576.https://doi.org/10.1137/110825078Google Scholar
  11. A. Chakib, I. Khalil, H. Ouaissa, A. Sadik: On an effective approach in shape optimization problem for Stokes equation, Optimization Letters 17/6 (2023) 1487–1494.https://doi.org/10.1007/s11590-023-01970-5Google Scholar
  12. A. Chakib, I. Khalil, H. Ouaissa, A. Sadik: On a numerical approach for solving some geometrical shape optimization problems in fluid mechanics, Comm. Nonlinear Sci. Numer. Simulation 128 (2024), art.no. 107651, 24p.https://doi.org/10.1016/j.cnsns.2023.107651Google Scholar
  13. A. Chakib, I. Khalil, A. Sadik: An improved numerical optimization approach for solving geometrical shape optimization problems, Numer. Algorithms 96/2 (2024) 621–663.https://doi.org/10.1007/s11075-023-01660-4Google Scholar
  14. F. H. Clarke, R. Stern, P. Wolenski: Proximal smoothness and the lower-C2 property, J. Convex Analysis 2 (1995) 117–144.https://doi.org/10.68381/jca02008Google Scholar
  15. A. Colesanti: Brunn-Minkowski inequalities for variational functionals and related problems, Advances Math. 194/1 (2005) 105–140.https://doi.org/10.1016/j.aim.2004.06.002Google Scholar
  16. A. Colesanti, M. Fimiani: The Minkowski problem for torsional rigidity, Indiana Univ. Math. J. 59/3 (2010) 1013–1040.https://doi.org/10.1512/iumj.2010.59.3937Google Scholar
  17. J. Dalphin: Some characterizations of a uniform ball property, ESAIM Proc. Surveys 45 (2014) 437–446.https://doi.org/10.1051/proc/201445045Google Scholar
  18. J. Dalphin: Uniform ball property and existence of optimal shapes for a wide class of geometric functionals, Interfaces Free Boundaries 20/2 (2018) 211–260.https://doi.org/10.4171/ifb/401Google Scholar
  19. M. C. Delfour, J. P. Zolésio: Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Advances in Design and Control Vol. 22, Society for Industrial and Applied Mathematics, Philadelphia (2011).https://doi.org/10.1137/1.9780898719826Google Scholar
  20. V. F. Demyanov, A. M. Rubinov: Quasidifferential Calculus. Optimization Software, Inc. Publications Division, New York (1986).https://doi.org/10.1007/BFb0121132Google Scholar
  21. V. F. Demyanov, A. M. Rubinov: Foundations of Nonsmooth Analysis and Quasidifferential Calculus (Russian), Optimization and Simulation Vol. 23, Nauka, Moscow (1990).Google Scholar
  22. J. I. Diaz, B. Kawohl: On convexity and star-shapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Analysis Appl. 177 (1993) 263–263.https://doi.org/10.1006/jmaa.1993.1257Google Scholar
  23. J. Droniou: Quelques Résultats sur les Espaces de Sobolev, Polycopié de l’Ecole Doctorale de Maths-Info de Marseille, Marseille (2001).Google Scholar
  24. L. Evans, R. Gariepy: Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992).Google Scholar
  25. H. Federer: Curvature measures, Trans. Amer. Math. Soc. 93/3 (1959) 418–491.https://doi.org/10.1090/S0002-9947-1959-0110078-1Google Scholar
  26. M. P. Friedlander, I. Macedo, T. K. Pong: Gauge optimization and duality, SIAM J. Optimization 24/4 (2014) 1999–2022.https://doi.org/10.1137/130940785Google Scholar
  27. D. Henrion, J. Malick: Projection methods in conic optimization, in: Handbook on Semidefinite, Conic and Polynomial Optimization, M. F. Anjos et al. (eds.), International Series in Operations Research and Management Science Vol. 166, Springer, New York (2011) 565–600.https://doi.org/10.1007/978-1-4614-0769-0_20Google Scholar
  28. L. Hoermander: Notions of Convexity, Progress in Mathematics Vol. 127, Birkhäuser, Basel (1994).Google Scholar
  29. R. B. Holmes: Smoothness of certain metric projections on Hilbert space, Trans. Amer. Math. Soc. 184 (1973) 87–100.https://doi.org/10.1090/S0002-9947-1973-0326252-2Google Scholar
  30. J. P. Ignizio, T. M. Cavalier: Linear Programming, Prentice-Hall, New York (1994).Google Scholar
  31. G. E. Ivanov: Weak convexity of sets and functions in a Banach space, J. Convex Analysis 22/2 (2015) 365–398.https://doi.org/10.68381/jca22017Google Scholar
  32. D. A. Jerison: The direct method in the calculus of variations for convex bodies, Advances Math. 122/2 (1996) 262–279.https://doi.org/10.1006/aima.1996.0062Google Scholar
  33. D. A. Jerison: A Minkowski problem for electrostatic capacity, Acta Math. 176/1 (1996) 1–47.https://doi.org/10.1007/BF02547334Google Scholar
  34. S. G. Krantz, H. R. Parks: On the vector sum of two convex sets in space, Canadian J. Math. 43/2 (1991) 347–355.https://doi.org/10.4153/CJM-1991-020-7Google Scholar
  35. A. A. Niftiyev, Y. S. Gasimov: Control by Boundaries and Eigenvalue Problems with Variable Domains (Russian), Publishing House of the Baku State University, Baku (2004).Google Scholar
  36. C. Nour, H. Saoud, J. Takche: Regularization via sets satisfying the interior sphere condition, J. Convex Analysis 25/1 (2018) 1–19.https://doi.org/10.68381/jca25001Google Scholar
  37. C. Nour, R. J. Stern, J. Takche: The union of uniform closed balls conjecture, Control Cybernetics 38 (2009) 1525–1534.Google Scholar
  38. C. Nour, R. J. Stern, J. Takche: Proximal smoothness and the exterior sphere condition, J. Convex Analysis 16 (2009) 501–514.https://doi.org/10.68381/jca16027Google Scholar
  39. C. Nour, R. J. Stern, J. Takche: The θ-exterior sphere condition, ϕ-convexity, and local semiconcavity, Nonlinear Analysis Theory, Methods Appl. 73/2 (2010) 573–589.https://doi.org/10.1016/j.na.2010.04.001Google Scholar
  40. C. Nour, R. J. Stern, J. Takche: Validity of the union of uniform closed balls conjecture, J. Convex Analysis 18 (2011) 589–600.https://doi.org/10.68381/jca18037Google Scholar
  41. H. Poincaré: Sur les équations aux dérivées partielles de la physique mathématique, Amer. J. Math. 12/3 (1890) 211–294.https://doi.org/10.2307/2369620Google Scholar
  42. R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000) 5231–5249.https://doi.org/10.1090/S0002-9947-00-02550-2Google Scholar
  43. A. M. Rubinov: Abstract Convexity and Global Optimization, Nonconvex Optimization and its Applications Vol. 44, Kluwer Publications, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9Google Scholar
  44. R. Schneider: Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press Vol. 151, Cambridge (2014).https://doi.org/10.1017/CBO9781139003858Google Scholar
  45. C. Thaele: 50 years sets with positive reach – a survey, Surveys Math. Appl. 3 (2008) 123–165.Google Scholar
  46. L. Thibault: Unilateral Variational Analysis in Banach Spaces. Part II: Special Classes of Functions and Sets, World Scientific, Singapore (2023).https://doi.org/10.1142/12797Google Scholar
  47. L. Webster: Convexity, Oxford University Press, Oxford (1994).https://doi.org/10.1093/oso/9780198531470.001.0001Google Scholar
  48. S. Yamanaka, N. Yamashita: Duality of optimization problems with gauge functions, Optimization 73/4 (2024) 1025–1055.https://doi.org/10.1080/02331934.2022.2138384Google Scholar

Suggested citation

A. Boulkhemair, A. Chakib, A. Sadik. “On Some Uniform Estimates of Gauge Functions with Respect to Domains.” Journal of Convex Analysis 32 (2025), No. 4, 961–974. https://doi.org/10.68381/jca32051

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.