The Güntürk-Thao theorem revisited: polyhedral cones and limiting examples

Abstract

In 2023, Güntürk and Thao proved that the sequence $(x^{(n)})_{n\in\mathbb{N}}$ generated by random (relaxed) projections drawn from a finite collection of innately regular closed subspaces in a real Hilbert space satisfies $\sum_{n\in\mathbb{N}} \|x^{(n)}-x^{(n+1)}\|^γ<+\infty$ for all $γ>0$. We 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 $\mathbb{R}^2$, and for a plane and a non-polyhedral cone in $\mathbb{R}^3$.

Figures (2)

• Figure 1: The trajectory of \ref{['251213b']} for $f(x)=\exp(- x^{-2})$ with domain $\bigl[-\sqrt{2/3},\sqrt{2/3}\bigr]$
• Figure 2: The trajectory of \ref{['251215b']} for $f(x)=\exp(- x^{-2})$ with domain $\bigl[-\sqrt{2/3},\sqrt{2/3}\bigr]$

Theorems & Definitions (24)

• proof
• proof
• Corollary 2.4
• proof
• Corollary 2.5
• proof
• Theorem 2.7: polyhedral cones in Euclidean space
• proof
• Theorem 3.2: main result
• proof