On Some Uniform Estimates of Gauge Functions with Respect to Domains
A. Boulkhemair, Abdelkrim Chakib, Azeddine Sadik · Journal of convex analysis · 2025
We establish uniform estimates and properties of gauge functions for domains \Omega_\varepsilon Ω ε , \varepsilon\in[0,1] ε ∈ [ 0 , 1 ] , defined by the Minkowski sum \Omega_\varepsilon=\Omega_0+\varepsilon\Omega Ω ε = Ω 0 + ε Ω where \Omega_0 Ω 0 and \Omega Ω are convex and bounded subsets of \mathbb{R}^n 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 \Omega_\varepsilon Ω ε 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 Ω ε and its gradient as well as its hessian, with respect to the parameter \varepsilon ε .