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We demonstrate that the set $L^\infty(X, [-1,1])$ of all measurable functionsover a Borel measure space $(X, \mathcal B, \mu )$ with values in the unitinterval is typically non-polyhedric when interpreted as a subset of a dualspace. Our findings contrast the classical result that subsets of Dirichletspaces with pointwise upper and lower bounds are polyhedric. In particular,additional structural assumptions are unavoidable when the concept ofpolyhedricity is used to study the differentiability properties of solutionmaps to variational inequalities of the second kind in, e.g., the spaces$H^{1/2}(\partial \Omega)$ or $H_0^1(\Omega)$.

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