On Some Uniform Estimates of Gauge Functions with Respect to Domains

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 $\varepsilon$.