论文标题
Grunbaum猜想的简单证明
A simple proof of the Grunbaum conjecture
论文作者
论文摘要
令$λ_\ mathbb {k}（m）$表示尺寸$ m $子空间上的最大绝对投影常数。除了$ m = 1 $的微不足道的情况外，唯一已知的已知值为$λ_\ mathbb {k}（m）$是$ m = 2 $和$ \ mathbb {k} = \ mathbb {r {r} $在1960年。 2010年，B。Chalmers和G. Lewicki证明了这一点。 2019年，G。Basso提供了这种猜想的替代证明。这两个证据都非常复杂，并且有一种强烈的信念，即在其他情况下为$λ_\ mathbb {k}（m）$提供确切的价值将是一项艰巨的任务。在我们的论文中，我们介绍了值$λ_\ Mathbb {k}（m）$的上限，这成为众多情况的确切值。关键将结合文章的一些结果[B. Bukh，C。Cox，几乎是正交的载体和小的反式球形代码，ISR。 J. Math。 238，359-388（2020）]和[G. Basso，最大投影常数的计算，J。Funct。肛门。 277/10（2019），3560-3585。]，为此提供简化的证明。
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.
