Journal of Convex Analysis 32 (2025), No. 4, 961--974
Copyright Heldermann Verlag 2025
On Some Uniform Estimates of Gauge Functions with Respect to Domains
Abdesslam Boulkhemair
Jean Leray Math. Lab., UFR-Sciences and Techniques, 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) Besancon Mathematics Laboratory, University of Franche-Comte, UMR 6623 CNRS UBFC, Besancon, France
sadik.fstbm@gmail.com
[Abstract-pdf]
We establish uniform estimates and properties of gauge functions for domains $\Omega_\varepsilon$, $\varepsilon\in[0,1]$, defined by the Minkowski sum $\Omega_\varepsilon=\Omega_0+\varepsilon\Omega$ where $\Omega_0$ and $\Omega$ are convex and bounded subsets of $\mathbb{R}^n$. 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 [{\it 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 $\Omega_\varepsilon$ 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 $\Omega_\varepsilon$ and its gradient as well as its hessian, with respect to the parameter $\varepsilon$.
Keywords: Convex domains, gauge functions, support functions, Minkowski sum, uniform ball condition, reach condition, uniform estimates.
MSC: 52A07; 52A30, 53A05, 49Q10.
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