Finite-codimensional subspaces of Daugavet spaces: projection constants and minimal projections

Over the real or complex field, we establish a duality formula for projection constants of finite-codimensional subspaces of Banach spaces with the Daugavet property. If $$Y=\bigcap _{j=1}^n\ker f_j\subset X,W=\mathrm{span}\{f_1,\dots ,f_n\}\subset X^{\ast },$$ then $$\lambda (Y,X)=1+\lambda (W,X^{\ast }),$$ and minimal projections onto $Y$ correspond exactly to weak${}^{\ast }$-continuous minimal projections onto $W$. This yields, in particular, a complete description of the hyperplane case: every hyperplane has projection constant $2$, and $\ker f$ admits a minimal projection if and only if $f$ attains its norm. We then specialise to the real space $X=C[0,1]$. Our second ingredient is a transfer principle from duplication-stable finite-dimensional subspaces of ${\mathrm{\ell }}_1^N$ to piecewise-constant subspaces of $L_1[0,1]\subset M[0,1]=C[0,1{]}^{\ast }$. For the regular symmetric spaces constructed by Chalmers and the second-named author and the second named author and Prophet, respectively, the transferred subspaces retain their projection constants but admit no weak${}^{\ast }$-continuous minimal projections. Passing to annihilators yields finite-codimensional subspaces of the real space $C[0,1]$ for which the infimum defining the projection constant is not attained. As a consequence, for every $\mathrm{\Lambda }\in [2,\mathrm{\infty })$ there exists a finite-codimensional subspace $Y$ of the real space $C[0,1]$ such that $$\lambda (Y,C[0,1])=\mathrm{\Lambda },$$ and the infimum defining $\lambda (Y,C[0,1])$ is not attained. For each even codimension $n$ we moreover realise every value in the interval $(2,1+{\beta }_n]$, where $${\beta }_n={\mathsf{E}}_{{\mathsf{P}}_n}\mid \sum _{j=1}^n{\epsilon }_j\mid =n{2}^{-n}\left(\genfrac{}{}{0px}{}{n}{n\mathrm{/}2}\right)\sim \sqrt{\frac{2n}{\pi }},$$ $({\epsilon }_j)$ is a Rademacher family on ${\mathrm{\Omega }}_n=\{-1,1{\}}^n$, and ${\mathsf{P}}_n$ is the uniform probability measure.