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It follows easily from a result of Lindenstrauss that, for any real twodimensional subspace v of L¹, the relative projection constant λ(v;L¹) of v equals its (absolute) projection constant $\lambda \left(v\right)=su{p}_X\lambda (v;X)$. The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in $L^{\infty }\left(\nu \right)$ and isometric to v and a projection $P_{\infty }$ from C ⊕ V onto V such that $\parallel P_{\infty }\parallel =\parallel P₁\parallel$, where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if $P₁={\sum }_{i=1}^2{U}_i\otimes v_i$, then $P_{\infty }={\sum }_{i=1}^2{u}_i\otimes V_i$, where $d{V}_i=2v_{i}d\nu$ and $d{U}_i=-2u_{i}d\nu$.