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

Abstract: 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, \qquad W=\operatorname{span}{f_1,\dots,f_n} \subset X^*, ] then [ λ(Y,X)=1+λ(W,X^*), ] and minimal projections onto $Y$ correspond exactly to weak$^*$-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 $\ell_1^N$ to piecewise-constant subspaces of $L_1[0,1]\subset M[0,1]=C[0,1]^*$. 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$^*$-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 $Λ\in[2,\infty)$ there exists a finite-codimensional subspace $Y$ of the real space $C[0,1]$ such that [ λ(Y,C[0,1])=Λ, ] and the infimum defining $λ(Y,C[0,1])$ is not attained. For each even codimension $n$ we moreover realise every value in the interval $(2,1+βn]$, where [ β_n = \mathsf E{{\mathsf P}n}\Bigl|\sum{j=1}^n \varepsilon_j\Bigr| = n2^{-n}\binom{n}{n/2} \sim \sqrt{\frac{2n}π}, ] $(\varepsilon_j)$ is a Rademacher family on $Ω_n={-1,1}^n$, and $\mathsf{P}_n$ is the uniform probability measure.

• The paper establishes a duality principle linking the projection constant of a finite-codimensional subspace in a Daugavet space with that of its finite-dimensional dual.
• It provides explicit constructions in C[0,1] demonstrating that every hyperplane has a projection constant of 2 and shows non-attainment phenomena for higher codimensions.
• The work employs combinatorial and geometric methods to detail conditions for weak*-continuous minimal projections and the precise range of attainable projection constants.

Projection Constants and Minimal Projections in Finite-Codimensional Subspaces of Daugavet Spaces

Introduction

This paper presents a comprehensive analysis of the projection constants and minimal projections of finite-codimensional subspaces within Banach spaces possessing the Daugavet property. The investigative focus is two-pronged: establishing a duality framework linking the geometric parameters of finite-codimensional subspaces to corresponding finite-dimensional dual spaces, and constructing explicit examples in $C[0,1]$ that demonstrate the precise range and non-attainment phenomena of projection constants.

Duality Principle for Projection Constants

The foundational contribution delineates a duality formula connecting the projection constant $\lambda(Y, X)$ of a finite-codimensional subspace $Y$—formed as an intersection of kernels of linearly independent functionals in $X^*$—with the corresponding projection constant $\lambda(W, X^*)$ of the finite-dimensional dual subspace $W$ spanned by these functionals. Specifically, for $Y = \bigcap_{j=1}^n \ker f_j \subset X$ and $$W = \operatorname{span}\{f_1, \dots, f_n\} \subset X^*$$, it is shown that

$\lambda(Y, X) = 1 + \lambda(W, X^*).$

A crucial equivalence is also established: minimal projections onto $Y$ in $\lambda(Y, X)$0 exist if and only if minimal projections onto $\lambda(Y, X)$1 in $\lambda(Y, X)$2 exist and are weak$\lambda(Y, X)$3-continuous. This characterization provides a reduction from an infinite-dimensional scenario to a finite-dimensional dual framework, with the weak$\lambda(Y, X)$4 continuity constraint encapsulating the full geometric behavior.

The duality is especially rigid within Daugavet spaces, owing to the norm identity $\lambda(Y, X)$5 for every finite-rank $\lambda(Y, X)$6. This yields a lower bound $\lambda(Y, X)$7 for every proper finite-codimensional subspace $\lambda(Y, X)$8, and completely resolves the hyperplane case: every hyperplane has projection constant $\lambda(Y, X)$9, and minimal projections exist if and only if the annihilating functional attains its norm.

Transfer Principle and Explicit Constructions in $Y$0

Specializing to $Y$1 and its dual $Y$2, the work advances a transfer mechanism from finite-dimensional, duplication-stable subspaces of $Y$3 to piecewise-constant subspaces of $Y$4, and subsequently to their annihilators in $Y$5. This transfer preserves projection constants and accurately controls weak$Y$6 non-attainment.

Key results include the construction of regular symmetric finite-dimensional subspaces (from the work of Chalmers, Lewicki, and collaborators) that are duplication-stable—i.e., their block-duplicates retain both the projection constant and the uniqueness of minimal projections. The transferred copies in $Y$7 inherit projection constants from the original spaces but admit no weak$Y$8-continuous minimal projections when $Y$9. For each even codimension $X^*$0, all values in the interval $X^*$1 are realized as projection constants of subspaces in $X^*$2 for which non-attainment holds, with $X^*$3 determined as the mean absolute displacement in a symmetric random walk.

Quantitative and Structural Results

Highlighted numerical phenomena include:

• For every $X^*$4, there exists a finite-codimensional subspace $X^*$5 with $X^*$6, and the infimum is not attained.
• Every hyperplane in $X^*$7 has $X^*$8, with non-attainment entirely characterized by norm-attainment of the annihilating functional.
• For each even $X^*$9, all projection constant values in $\lambda(W, X^*)$0 are realized, where $\lambda(W, X^*)$1, providing asymptotic tightness up to the universal factor $\lambda(W, X^*)$2 with respect to the Kadec–Snobar bound ($\lambda(W, X^*)$3).

These results rely predominantly on the geometric duality developed for Daugavet spaces and the explicit combinatorial structure of regular symmetric subspaces. The methodology leverages the absence of weak$\lambda(W, X^*)$4-continuous minimal projections in infinite-dimensional, yet highly structured, subspaces.

Non-attainment Phenomena and Theoretical Implications

The examples constructed show that for a broad range of projection constants (dense in $\lambda(W, X^*)$5), the infimum norm of projections is not achieved—a sharp distinction from classical finite-dimensional behavior. This extends non-attainment phenomena into a highly geometric context, intimately connected with the Daugavet property.

The results also tie the existence of minimal projections to extrinsic norm-attainment properties in the dual and underline a precise dichotomy: while every hyperplane in $\lambda(W, X^*)$6 has projection constant two, only those annihilated by norm-attaining functionals admit minimal projections.

Moreover, the explicit realization of codimension–projection constant pairs demonstrates the fine structure and limitations of subspace geometry in Daugavet settings. The paper positions these findings alongside classical questions on absolute projection constants, maximal projection constants, and the precise values attainable in $\lambda(W, X^*)$7 and related dual Daugavet spaces.

Open Problems and Future Directions

Unresolved questions include the full characterization of the possible values of projection constants for codimension-$\lambda(W, X^*)$8 subspaces in $\lambda(W, X^*)$9 without minimal projections—specifically, whether every value in $W$0 (with $W$1 the maximal $W$2-dimensional absolute projection constant) is attainable. The challenge at the endpoint $W$3 lies in realizing annihilators that are both extremal for projection constants and preclude weak$W$4-continuous minimal projections.

From a theoretical perspective, the framework and transfer techniques are poised for generalization to other Daugavet spaces, including $W$5 for perfect compacta or duals of $W$6-spaces. Furthermore, the relationship between combinatorial constructions (such as those producing explicit regular symmetric subspaces) and asymptotically optimal non-attainment families remains a fertile ground for exploration.

Conclusion

This work delivers a rigorous classification of projection constants and minimal projections for finite-codimensional subspaces in Banach spaces with the Daugavet property, with a specialized focus on $W$7. The established duality reduces infinite-dimensional projection problems to finite-dimensional dual computations subject to weak$W$8 continuity, facilitating explicit and asymptotically optimal non-attaining constructions. These results not only resolve classical questions but also set clear lines of inquiry for the geometry and combinatorics of Banach space subspaces, with broader implications for the structure of function spaces and dual continuous operators.