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K-sets and the K-number of real sup-norm spaces

Abstract

For real sup-norm spaces of dimension n ≥ 2, an extreme-point set is a K-set exactly when it spans, and the K-number is n.

Sain, Manna and Paul, arXiv:2407.08900v1, Definition 1: “Definition 1. Let X be a Banach space. A set A ⊂ S_X is said to be a Koldobsky-set, for the sake of brevity, a K-set, if any operator T ∈ L(X) that preserves Birkhoff-James orthogonality at each point of A, is necessarily a scalar multiple of an isometry.”

Definition 3 of the same paper: “Definition 3. Let X be a Banach space having a finite K-set. Then the K-number of X is defined by min{|A| : A is a K-set of X}. We denote the K-number of X by κ(X).”

The Open Question at the end of §2 reads: “Open Question: Characterize the minimal K-sets in ℓ^n_∞ and hence find κ(ℓ^n_∞).” Minimal means minimal under inclusion. The characterization below concerns extreme-point sets; minimal K-sets containing non-extreme points are not classified.

Definition 1.1 (Birkhoff–James orthogonality).

Formalization. D5/S3/Analytic/BanachGeometry/LinfKNumber.BJOrth (✓ std3).

Citation. D. Sain; J. Manna; K. Paul (2024). On local preservation of orthogonality and its application to isometries. URL: https://arxiv.org/abs/2407.08900v1.

Commentary.

For a real seminormed space E, BJOrth x y requires that adding any real multiple of y cannot decrease the norm of x.

Definition 1.2 (Preservation at a point).

Formalization. D5/S3/Analytic/BanachGeometry/LinfKNumber.PreservesBJAt (✓ std3).

Citation. D. Sain; J. Manna; K. Paul (2024). On local preservation of orthogonality and its application to isometries. URL: https://arxiv.org/abs/2407.08900v1.

Commentary.

PreservesBJAt T x means that for every y, x ⊥_B y ⇒ Tx ⊥_B Ty. The operator is real linear.

Definition 1.3 (A scalar multiple of an isometry).

Formalization. D5/S3/Analytic/BanachGeometry/LinfKNumber.IsScalarMulIsometry (✓ std3).

Citation. D. Sain; J. Manna; K. Paul (2024). On local preservation of orthogonality and its application to isometries. URL: https://arxiv.org/abs/2407.08900v1.

Commentary.

The predicate states that one nonnegative real scalar c satisfies ‖T y‖ = c ‖y‖ for every y. It includes T = 0. For c > 0 on the finite-dimensional sup-norm carrier, dividing T by c gives a linear isometry; all linear endomorphisms of that carrier are bounded.

Definition 1.4 (Koldobsky sets).

Formalization. D5/S3/Analytic/BanachGeometry/LinfKNumber.IsKSet (✓ std3).

Citation. D. Sain; J. Manna; K. Paul (2024). On local preservation of orthogonality and its application to isometries. URL: https://arxiv.org/abs/2407.08900v1.

Commentary.

IsKSet A requires A to lie on the unit sphere and every real linear operator preserving Birkhoff–James orthogonality at every point of A to scale the norm. In the real finite-dimensional specialization this encodes Definition 1.

Definition 1.5 (The extreme-point characterization and cardinality bounds).

Formalization. D5/S3/Analytic/BanachGeometry/LinfKNumber.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. D. Sain; J. Manna; K. Paul (2024). On local preservation of orthogonality and its application to isometries. URL: https://arxiv.org/abs/2407.08900v1.

Commentary.

Let X be Fin n → ℝ with the sup norm, V the extreme points of its closed unit ball, and n ≥ 2. The three conjuncts assert that every A ⊆ V is a K-set exactly when its real span is X, that an n-element K-set exists, and that every finite K-set has at least n elements. Together the last two give κ(X) = n under Definition 3. The extreme points are exactly the vectors with every coordinate of absolute value one. Their inclusion-minimal spanning subsets are exactly the bases contained in {±1}ⁿ.

Theorem 1.6 (Strict descent at active coordinates).

Proof. Machine-checked in Lean as D5/S3/Analytic/BanachGeometry/LinfKNumber.active_not_BJ (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. D. Sain; J. Manna; K. Paul (2024). On local preservation of orthogonality and its application to isometries. URL: https://arxiv.org/abs/2407.08900v1.

Commentary.

For a nonzero finite real coordinate vector w, if zᵢwᵢ is strictly positive at every coordinate where |wᵢ| = ‖w‖, a sufficiently small negative multiple of z decreases every coordinate below ‖w‖. Finiteness gives one common step size. Therefore w is not Birkhoff–James orthogonal to z. This criterion controls the active facets in the image-vertex proof and in the polyhedral plane counterexample.

Theorem 1.7 (Proof of the characterization and the K-number).

Proof. Machine-checked in Lean as D5/S3/Analytic/BanachGeometry/LinfKNumber.result (✓ std3). ∎

Resolves. Problems/sain-manna-paul-2024-linf-k-number (proved) by D5/S3/Analytic/BanachGeometry/LinfKNumber.result.

Source. Repository-derived.

Acknowledgement. D. Sain; J. Manna; K. Paul (2024). On local preservation of orthogonality and its application to isometries. URL: https://arxiv.org/abs/2407.08900v1.

Commentary.

At a vertex u, a zero coordinate of y implies u ⊥B y. A kernel vector contradicts preservation at a vertex with nonzero image, so a nonzero operator preserving at a spanning vertex set is injective. The active-coordinate inequality forces every coordinate of T u to have absolute value ‖T u‖. For S = T⁻¹, the proof gives uᵢ Sᵢⱼ (T u)ⱼ ≥ 0. Two nonzero entries in one column would constrain all test vertices to uᵢ = ±uₖ, contradicting span. Thus S has permutation support, and its action on one vertex fixes the common absolute value of its nonzero entries. This proves norm scaling. Conversely a nonzero rank-one operator annihilating a proper span preserves at that set but cannot scale the norm in dimension at least two. Finally, the all-ones vector and the n − 1 vectors obtained by changing one nonzero-index coordinate to −1 form an n-element spanning vertex set. The characterization also implies that any n linearly independent extreme points suffice in these sup-norm spaces.

References