发表机构
Institute of Mathematics University of the National Education Commission; Faculty of Mathematics and Computer Science Jagiellonian University（国家教育委员会大学数学研究所; 雅盖隆大学数学与计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究围绕巴拿赫空间中极大绝对投影常数的求解难题，分析双角紧框架、正单纯形边中点等结构化向量构型的性质，结合数值证据提出相关猜想，为该领域研究提供新方向。
AI 中文摘要
设$\lambda_{\mathbb K}(m)$表示数域$\mathbb K=\mathbb R$或$\mathbb C$上$m$维巴拿赫空间中的极大绝对投影常数。目前仅在少数情形下已知其精确值，确定该值仍是一个具有挑战性的问题。在本注记中，我们研究了在此背景下自然出现的若干结构化向量构型。我们首先回顾极大投影常数与紧框架之间的联系，随后考虑双角紧框架，并证明其相对投影常数与拟极大投影常数相等。正单纯形的边中点提供了一个尤为有趣的示例，它们给出了$\lambda_{\mathbb R}(m)$的自然下界。数值证据表明，这些下界在维度6和8时可能是紧的。我们还讨论了含少量向量的加权球面$(2,2)$-设计及其与极大投影常数的联系。维度4和5下的示例表明，其格拉姆矩阵的符号模式可能发挥重要作用。这些观察结果促使我们提出了若干关于极大绝对投影常数及相关向量构型的猜想。
英文摘要
Let $λ_{\mathbb K}(m)$ 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 $λ_{\mathbb R}(m)$. 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.
