A proof of the Grünbaum conjecture

Tom 200 / 2010

Bruce L. Chalmers, Grzegorz Lewicki Studia Mathematica 200 (2010), 103-129 MSC: Primary 46A22, 47A30; Secondary 47A58. DOI: 10.4064/sm200-2-1


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


  • Bruce L. ChalmersDepartment of Mathematics
    University of California
    Riverside, CA 92521, U.S.A.
  • Grzegorz LewickiInstitute of Mathematics
    Jagiellonian University
    30-348 Kraków, Poland

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