The Güntürk–Thao theorem revisited: Polyhedral cones and limiting examples

In 2023, Güntürk and Thao proved that the sequence ( 𝑥 ( 𝑛 ) ) 𝑛 ∈ ℕ generated by random (relaxed) projections drawn from a finite innately regular collection of closed subspaces in a real Hilbert space satisfies ∑ 𝑛 ∈ ℕ ‖ 𝑥 ( 𝑛 ) − 𝑥 ( 𝑛 + 1 ) ‖ 𝛾 < + ∞ for all 𝛾 > 0 . We show that the Güntürk–Thao result is sharp in the sense that it fails for two closed subspaces that is not innately regular. We also extend their result to a finite collection of polyhedral cones. Moreover, we construct examples showing the tightness of our extension: indeed, the result fails for a line and a convex set in ℝ 2 , and for a plane and a nonpolyhedral cone in ℝ 3 .