A simple proof of the Grunbaum conjecture
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| Format: | Preprint |
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2022
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| author | Deregowska, Beata Lewandowska, Barbara |
| author_facet | Deregowska, Beata Lewandowska, Barbara |
| contents | Let $λ_\mathbb{K}(m)$ denote the maximal absolute projection constant over the subspaces of dimension $m$. Apart from the trivial case for $ m=1$, the only known value of $λ_\mathbb{K}(m)$ is for $ m=2$ and $\mathbb{K}=\mathbb{R}.$ In 1960, B.Grunbaum conjectured that $λ_\mathbb{R}(2)=\frac{4}{3}$ and in 2010, B. Chalmers and G. Lewicki proved it. In 2019, G. Basso delivered the alternative proof of this conjecture. Both proofs are quite complicated, and there was a strong belief that providing an exact value for $λ_\mathbb{K}(m)$ in other cases will be a tough task. In our paper, we present an upper bound of the value $λ_\mathbb{K}(m)$, which becomes an exact value for the numerous cases. The crucial will be combining some results from the articles [B. Bukh, C. Cox, Nearly orthogonal vectors and small antipodal spherical codes, Isr. J. Math. 238, 359-388 (2020)] and [G. Basso, Computation of maximal projection constants, J. Funct. Anal. 277/10 (2019), 3560-3585.], for which simplified proofs will be given. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_09454 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A simple proof of the Grunbaum conjecture Deregowska, Beata Lewandowska, Barbara Functional Analysis 41A65, 41A44, 46B20, 15A42, 42C15 Let $λ_\mathbb{K}(m)$ denote the maximal absolute projection constant over the subspaces of dimension $m$. Apart from the trivial case for $ m=1$, the only known value of $λ_\mathbb{K}(m)$ is for $ m=2$ and $\mathbb{K}=\mathbb{R}.$ In 1960, B.Grunbaum conjectured that $λ_\mathbb{R}(2)=\frac{4}{3}$ and in 2010, B. Chalmers and G. Lewicki proved it. In 2019, G. Basso delivered the alternative proof of this conjecture. Both proofs are quite complicated, and there was a strong belief that providing an exact value for $λ_\mathbb{K}(m)$ in other cases will be a tough task. In our paper, we present an upper bound of the value $λ_\mathbb{K}(m)$, which becomes an exact value for the numerous cases. The crucial will be combining some results from the articles [B. Bukh, C. Cox, Nearly orthogonal vectors and small antipodal spherical codes, Isr. J. Math. 238, 359-388 (2020)] and [G. Basso, Computation of maximal projection constants, J. Funct. Anal. 277/10 (2019), 3560-3585.], for which simplified proofs will be given. |
| title | A simple proof of the Grunbaum conjecture |
| topic | Functional Analysis 41A65, 41A44, 46B20, 15A42, 42C15 |
| url | https://arxiv.org/abs/2206.09454 |