  We demonstrate that the set $L^\infty(X, [-1,1])$ of all measurable functions
over a Borel measure space $(X, \mathcal B, \mu )$ 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
$H^{1/2}(\partial \Omega)$ or $H_0^1(\Omega)$.

Optimization and Control