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

TL;DR

This paper shows that the set of bounded measurable functions with values in [-1,1] in a dual space is generally non-polyhedric, contrasting with classical polyhedricity results for Dirichlet spaces, impacting variational inequality analysis.

Contribution

It demonstrates the non-polyhedricity of certain function sets in dual spaces, highlighting the need for structural assumptions in variational inequality studies.

Findings

Sets in dual spaces are typically non-polyhedric.

Classical polyhedricity results do not extend to these function sets.

Additional assumptions are necessary for differentiability analysis.

Abstract

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 $H_0^1(\Omega)$.