arXiv subjects

Barbara Lewandowska

Publications and source records attributed to Barbara Lewandowska.

4 recordsLinked to original sources

arXiv2026

Maximal Projection Constants and Extremal Vector Configurations: Some Conjectures and Examples

Let $\lambda_{\mathbb K}(m)$ denote the maximal absolute projection constant among $m$-dimensional Banach spaces over $\mathbb K=\mathbb R$ or $\mathbb C$. Its exact value is known only in a few cases, and determining it remains a challenging problem. In this note, we investigate several structured vector configurations that naturally arise in this context. We first recall the connection between maximal projection constants and tight frames, then consider biangular tight frames, and show that their relative and quasimaximal projection constants coincide. A particularly interesting example is provided by the midpoints of the edges of a regular simplex, which yield natural lower bounds for $\lambda_{\mathbb R}(m)$. Numerical evidence suggests that these bounds may be sharp in dimensions $6$ and $8$. We also discuss weighted spherical $(2,2)$-designs with a small number of vectors and their connection with maximal projection constants. Examples in dimensions $4$ and $5$ indicate that the sign patterns of their Gram matrices may play an important role. These observations lead us to formulate several conjectures concerning maximal absolute projection constants and the vector configurations associated with them.

math.FA↗

arXiv2025

When is a subspace of $\ell_\infty^N$ isometrically isomorphic to $\ell_\infty^n$?

It is shown in this note that one can decide whether an $n$-dimensional subspace of $\ell_\infty^N$ is isometrically isomorphic to $\ell_\infty^n$ by testing a finite number of determinental inequalities. As a byproduct, an elementary proof is provided for the fact that an $n$-dimensional subspace of $\ell_\infty^N$ with projection constant equal to one must be isometrically isomorphic to $\ell_\infty^n$.

math.FA↗

arXiv2024

A simple proof of the Grunbaum conjecture

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.

math.FA↗

arXiv2022

On the value of the fifth maximal projection constant

Let $λ(m)$ denote the maximal absolute projection constant over real $m$-dimensional subspaces. This quantity is extremely hard to determine exactly, as testified by the fact that the only known value of $λ(m)$ for $m>1$ is $λ(2)=4/3$. There is also numerical evidence indicating that $λ(3)=(1+\sqrt{5})/2$. In this paper, relying on a new construction of certain mutually unbiased equiangular tight frames, we show that $λ(5)\geq 5(11+6\sqrt{5})/59 \approx 2.06919$. This value coincides with the numerical estimation of $λ(5)$ obtained by B. L. Chalmers, thus reinforcing the belief that this is the exact value of $λ(5)$.

math.FA↗