On the Non‐Polyhedricity of Sets with Upper and Lower Bounds in Dual Spaces

Abstract

We demonstrate that the set L∞(X, [−1,1]) of all measurable functions over a Borel measure space (X, B, μ) with values in the unit interval is typically non‐polyhedric when interpreted as a subset of a dual space. Our findings contrast the classical result that subsets of Dirichlet spaces with pointwise upper and lower bounds are polyhedric. In particular, additional structural assumptions are unavoidable when the concept of polyhedricity is used to study the differentiability properties of solution maps to variational inequalities of the second kind in, e.g., the spaces H1/2(∂Ω) or H01(Ω).

Strong Stationarity for Optimal Control of Variational Inequalities of the Second Kind

“…is in general not polyhedric, as the counterexamples in [5,6,28] demonstrate. If one assumes that this set is polyhedric, then the analysis of [12,19] can be adapted to prove the directional differentiability of S, see [17,23].…”

Optimal control of elliptic variational inequalities with bounded and unbounded operators

“…(Ω)} is not polyhedric in H −1 (Ω) (see the recent contribution [9]). Thus, additional assumptions have to be imposed in [11] to guarantee the polyhedricity.…”

“…The idea is to make use of the polyhedricity of the test set ∂ϕ(0) appearing in the dual formulation of the constraint in (P b ). By means of [9,Thm. 2.3], this property allows us to derive the directional differentiability of the solution operator, which leads to a strong stationary optimality system.…”

On the optimal control of some nonsmooth distributed parameter systems arising in mechanics

“…An alternative polyhedricity hypothesis has been considered recently in [43]. Although the latter apparently avoids structural assumptions on the active set, in the recent work [19] it is shown that in order to get polyhedricity, structural assumptions on the active set are actually unavoidable. Theorem 3.6 Let f, h ∈ L r (Ω) with r > max{d/2, 1} be given.…”