The Güntürk-Thao theorem revisited: polyhedral cones and limiting examples
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908759044915200 |
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| author | Bauschke, Heinz H. Tung, Tran Thanh |
| author_facet | Bauschke, Heinz H. Tung, Tran Thanh |
| contents | 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 $\mathbb{R}^3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_07002 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Güntürk-Thao theorem revisited: polyhedral cones and limiting examples Bauschke, Heinz H. Tung, Tran Thanh Optimization and Control 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 $\mathbb{R}^3$. |
| title | The Güntürk-Thao theorem revisited: polyhedral cones and limiting examples |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2601.07002 |