On Some Uniform Estimates of Gauge Functions with Respect to Domains

We establish uniform estimates and properties of gauge functions for domains

,

\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

.