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.
Highlights
1.1 Fixed points of compositions of projectors Throughout,X is a real Hilbert space (1) andC1, . . . , Cm are nonempty closed convex subsets of X, (2)© Salihah Alwadani & Heinz H
Which turns out to be independent of the cycle chosen, makes (4) true and yields “one half” of (5), namely: Fm ⊆ Cm ∩(Cm−1 +vm−1)∩· · ·∩(C1 +v1 +· · ·+vm−1) and analogously for Fm−1, . . . , F1. This description is not fully satisfying — it is only implicit in the sense it was not known what the difference vectors are when the fixed point sets Fi are empty
The resolution depends on key results from monotone operator theory and yields a formula for the difference vectors
Summary
This description is not fully satisfying — it is only implicit in the sense it was not known what the difference vectors are when the fixed point sets Fi are empty. Note that this description is not based on the fixed point sets F1, F2. (See [6, 7, 8, 17] for much more on the case when m = 2.) A referee pointed out that when m = 2 and F1 = F2 = ∅ one cannot expect uniqueness of the difference vectors as one may separate the sets even further. For results on underrelaxed projectors, see [5, 19].) Even when cycles exist, the “meaning” of the distance vector was not understood
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