A simple proof of the Grunbaum conjecture

TL;DR

This paper introduces a new upper bound for the maximal absolute projection constant in subspaces, providing exact values in many cases, and simplifies existing proofs related to the Grunbaum conjecture.

Contribution

It offers a simplified proof and an upper bound for the projection constant, extending known results and solving cases previously considered difficult.

Findings

Provides an exact value for many cases of the projection constant

Simplifies proofs of previous results

Introduces an upper bound that matches known values in several instances

Abstract

Let $\lambda_\mathbb{K}(m)$ 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 $\lambda_\mathbb{K}(m)$ is for $ m=2$ and $\mathbb{K}=\mathbb{R}.$ In 1960, B.Grunbaum conjectured that $\lambda_\mathbb{R}(2)=\frac{4}{3}$ 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 $\lambda_\mathbb{K}(m)$ in other cases will be a tough task. In our paper, we present an upper bound of the value $\lambda_\mathbb{K}(m)$, 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.