A note on Banach--Mazur problem

Abstract

We prove that if $X$ is a real Banach space, with $\dim X\ge 3$, which contains a subspace of codimension 1 which is 1-complemented in $X$ and whose group of isometries is almost transitive then $X$ is isometric to a Hilbert space. This partially answers the Banach-Mazur rotation problem and generalizes some recent related results.

8 pages, 2 figures but one of the figures doesn't run well in TeX so it is not included here. The ps file of this paper which includes all figures is available at http://www.users.muohio.edu/randrib/bm3.ps. to appear in Glasgow J. Math. (2002)