2025/05/30 by Tomasz Kobos, Kobos, Tomasz · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2505.24526
openalex publication_date 2025/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let n ≥ 2 be an integer such that an equiangular set of vectors w1, …, wd of the maximal possible cardinality (in relation to the the general Gerzon upper bound) exists in \mathbbKn, where \mathbbK=ℝ or \mathbbK=ℂ (i.e. d=(n(n+1))/(2) in the real and d=n2 in the complex case). We provide a complete characterization of n-dimensional normed spaces X having a maximal absolute projection constant among all n-dimensional normed spaces over \mathbbK. The characterization states that X has a maximal projection constant if and only if it is isometric to a space, for which the unit ball of the dual space is contained between the absolutely convex hull of the vectors w1, …, wd and an appropriately rescaled zonotope generated by the same vectors. As a consequence, we obtain that in the considered situations, the case of n=2 and \mathbbK=ℝ is the only one, where there is a unique norm in \mathbbKn (up to an isometry) with the maximal projection constant. In this case, the unit ball is an affine regular hexagon in ℝ2.