The difference vectors for convex sets and a resolution of the geometry conjecture

Abstract

The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets.In this paper, we provide a complete resolution of the geometry conjecture. Our proof relies on monotone operator theory. We revisit previously known results and provide various illustrative examples. Comments on the numerical computation of the quantities involved are also presented.

$m$th roots of the identity operator and the geometry conjecture

“…In Lemma 16, we give a result very similar to Lemma 6 using three (nontrivial) results on maximally monotone operators on a Hilbert space. This corresponds more closely with the method used in [3], and is what we call the maximally monotone operator approach.…”

“…This conjecture was finally solved in the affirmative in [3,Theorem 9,. In this paper, we give a proof of this conjecture which is simpler than that in [3], and extends the result to a more general situation. (See Theorem 7.)…”

Attouch-Théra Duality, Generalized Cycles and Gap Vectors

“…Remark 7.4. In this regard, see [2] for finding cycles and gap vectors of compositions of projections, and also [8,Section 3.3.3] for an abstract framework.…”

“…Many authors have studied cycles of compositions of proximal mappings or resolvents; see, e.g., [2,4,5,7,9,14,15,10,21]. For the compositions of two proximal mappings or resolvents, the investigation has matured; see [8,21].…”

“…On the one hand, a systematic study of cycles and gap vectors for compositions of proximal mappings does not exist in the literature; on the other hand, it is not clear what one should do when the cycles and gap vectors do not exist. In [2], we carried out the study of extended gap vectors for projection mappings and provided an answer to the geometry conjecture, which concerns the situation where the classical gap vector does not necessarily exist.…”