This note corrects a mistake in [3, section 3.2.1], concerning the characterization of the domain of an operator of the form Φ(t, •) := (∂σ Γ(t) + D) -1 (•), D = D ⊤ a constant positive semi-definite matrix, Γ(t) a nonempty closed convex set for each t, σ Γ(t) (•) its support function.

The results in [2, Theorems 3, 4] are wrongly used in [3] to conclude that dom(Φ(t, •)) = Γ(t) + Im(D), while it can only be inferred that dom(Φ(t, •)) ≃ Γ(t) + Im(D), where ≃ has the meaning as stated in [2]. Actually, the equality holds if in addition Γ(t) is polyhedral. The set Γ(t) is defined as

this is assumed in the subsequent developments in [3] hence it has no consequence on the results which are stated after section 3.2.1 "Calculation of sets" in that article). We will also use the following. If S := {γ ∈ IR m : Aγ ≥ b} for some A ∈ IR n×m and b ∈ IR n , then for any x ∈ S, the normal cone to S at x is given by:

(1)

, where ∂ denotes the subdifferential of convex analysis. Let D ≽ 0 (not necessarily symmetric), then SOL 0, GDG ⊤ is a polyhedral convex cone [5, p.1054], where SOL 0, GDG ⊤ is the set of solutions to the homogeneous Linear Complementarity Problem (LCP): 0 ≤ λ ⊥ GDG ⊤ λ ≥ 0. Therefore, there exists a positive number k and a matrix Ĝ ∈ IR k×j such that SOL 0,

+ (for instance, if GDG ⊤ ≻ 0, then k can be chosen as any positive number, and Ĝ ∈ IR k×j is a zero matrix since the only solution is null, while if GDG ⊤ = 0 then k = j can be chosen and Ĝ is an identity matrix since all vectors in IR j + are solutions). Let us remind that Φ(t, •) := (∂σ Γ(t) + D) -1 (•). The following result holds:

Lemma 1 Let H := h ∈ IR m : ĜGh ≥ 0 . Then, dom(Φ(t, •)) = H -F (t) for all t ≥ 0.

Proof: For each t ≥ 0, we have dom(Φ(t, •)) = Im N -1 K ⋆ -F (t) + D . Let us prove that