A proof of the Grünbaum conjecture

Bruce L. Chalmers ¹   ; Grzegorz Lewicki ²

¹ Department of Mathematics University of California Riverside, CA 92521, U.S.A.
² Institute of Mathematics Jagiellonian University 30-348 Kraków, Poland

Studia Mathematica, Tome 200 (2010) no. 2, pp. 103-129

Résumé

Let $V$ be an $n$-dimensional real Banach space and let $\lambda(V)$ denote its absolute projection constant. For any $N \in \Bbb{N}$ with $ N \geq n$ define $$\eqalign{ \lambda_n^N = \sup\{ \lambda(V): \dim(V)= n,\, V \subset l_{\infty}^{(N)} \},\cr \lambda_n = \sup\{ \lambda(V): \dim(V)= n \}. \cr}$$ A well-known Grünbaum conjecture [Trans. Amer. Math. Soc. 95 (1960)] says that $$ \lambda_2 = 4/3. $$ König and Tomczak-Jaegermann [J. Funct. Anal. 119 (1994)] made an attempt to prove this conjecture. Unfortunately, their Proposition 3.1, used in the proof, is incorrect. In this paper a complete proof of the Grünbaum conjecture is presented

Détail

DOI : 10.4064/sm200-2-1

Mots-clés : n dimensional real banach space lambda denote its absolute projection constant bbb geq define eqalign lambda sup lambda dim subset infty lambda sup lambda dim well known nbaum conjecture trans amer math soc says lambda nig tomczak jaegermann funct anal made attempt prove conjecture unfortunately their proposition proof incorrect paper complete proof nbaum conjecture presented