Let $X$ be a $d$-dimensional normed space and $M$ a subspace of $X$, then we say that $M$ is 1-complemented if there exists a linear projection $P :X \rightarrow X$ such that $P(X)=M$ and $\|P\|=1$.
What I wish to know is whether there always exists a $(d-1)$-dimensional subspace that is 1-complemented. It is a known result by Kakutani that $X$ is Euclidean if and only if all subspaces are 1-complemented (in fact we only require that all 2D subspaces be 1-complemented) and so in some way these spaces are special. If anyone knows of any such results I would be much appreciative.