dorsal/arxiv
View SchemaThe G\"unt\"urk-Thao theorem revisited: polyhedral cones and limiting examples
| Authors | Heinz H. Bauschke, Tran Thanh Tung |
|---|---|
| Categories | |
| ArXiv ID | 2601.07002vv1 |
| URL | https://arxiv.org/abs/2601.07002 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
In 2023, G\"unt\"urk 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 $\sum_{n\in\mathbb{N}} \|x^{(n)}-x^{(n+1)}\|^\gamma <+\infty$ for all $\gamma>0$. 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 $\mathbb{R}^3$.
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"date_created": "2026-02-17T05:53:08.659000Z",
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"abstract": "In 2023, G\\\"unt\\\"urk 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 $\\sum_{n\\in\\mathbb{N}} \\|x^{(n)}-x^{(n+1)}\\|^\\gamma \u003c+\\infty$ for all $\\gamma\u003e0$.\n 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 $\\mathbb{R}^3$.",
"arxiv_id": "2601.07002",
"authors": [
"Heinz H. Bauschke",
"Tran Thanh Tung"
],
"categories": [
"math.OC"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "The G\\\"unt\\\"urk-Thao theorem revisited: polyhedral cones and limiting examples",
"url": "https://arxiv.org/abs/2601.07002",
"version": "v1"
},
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