A Counterexample and Fix to a Minimum Distance Duality Theorem

TL;DR

This paper identifies a flaw in a classical duality theorem related to minimum distance problems, provides a counterexample, and proposes a corrected condition to restore the theorem's validity under standard definitions.

Contribution

The paper uncovers a false assumption in a well-known duality theorem and introduces a necessary condition to fix it within standard functional analysis frameworks.

Findings

Counterexample disproves the original theorem under standard definitions

A necessary condition involving annihilators restores the theorem's validity

Discussion on the limitations of Hahn-Banach extensions in this context

Abstract

We consider dual optimization problems to the fundamental problem of finding the minimum distance from a point to a subspace. We provide a counterexample to a theorem which has appeared in the literature, relating the minimum distance problem to a maximization problem in the predual space. The theorem was stated in a series of papers by Zames and Owen in the early 1990s in conjunction with a non-standard definition, which together are true, but the theorem is false when assuming standard definitions, as it would later appear. Reasons for the failure of this theorem are discussed; in particular, the fact that the Hahn-Banach Theorem cannot be guaranteed to provide an extension which is an element of the predual space. The condition needed to restore the theorem is derived; namely, that the annihilator of the pre-annihilator return the original subspace of interest. This condition is consistent with the non-standard definition initially used, and it is further shown to be necessary in a sense.