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.

Keywords

• Attouch–Théra duality
• circular right shift operator
• convex sets
• cycle
• fixed point set
• monotone operator theory
• projectors.