Metric Double Complements of Convex Sets
Douglas S. Bridges
TL;DR
The paper investigates how the metric double complement $-(-K)$ of a convex set $K$ in a normed space relates to the logical double negation $\lnot\lnot K$ and to the interior $K^{\circ}$ within Bishop's constructive mathematics. It proves that, when $K$ has inhabited interior, $-(-K)=(\lnot\lnot K)^{\circ}$, and in finite dimensions this interior condition can be dropped; additionally, if $K$ is located and either $K^{\circ}$ is inhabited or the space is finite-dimensional, then $-(-K)=K^{\circ}$. The work further provides simplex-based constructive proofs in finite dimensions and shows that these not-not and interior relations are dense and convex under suitable hypotheses, while also establishing that certain equalities cannot be strengthened without invoking non-constructive principles. Overall, the results clarify the constructive interplay between metric and logical notions for convex sets, delineating precise conditions under which metric and interior notions coincide and where inherent logical limitations arise.
Abstract
In constructive mathematics the metric complement of a subset S of a metric space X is the set -S of points in X that are bounded away from S. In this note we discuss, within Bishop's constructive mathematics, the connection between the metric double complement, -(-K), and the logical double complement, not not K, where K is a convex subset of a normed linear space X. In particular, we prove that if K has inhabited interior, then -(-K) equals the interior of not not K, that the hypothesis of inhabited interior can be dropped in the finite-dimensional case, and that we cannot constructively replace the interior of not not K by that of K in these results.