Gordon-Lewis Constant in Banach Spaces

• Gordon-Lewis constant is a quantitative measure that assesses whether 1-summing operators admit controlled 1-factorizations in local Banach space theory.
• It serves as a bridge between projection, unconditional basis, and Sidon constants, providing actionable bounds in analytic polynomial and Boolean cube spaces.
• In its geometric interpretation, the constant G_L(X) arises from inscribed triangle configurations, linking convexity, uniform non-squareness, and super-reflexivity.

Searching arXiv for recent and foundational papers on the Gordon-Lewis constant. The term Gordon-Lewis constant appears in two distinct senses in the Banach-space literature represented here. In the local theory of Banach spaces it denotes the invariant $\boldsymbol{gl}(X)$, defined as the smallest $C>0$ such that every 1-summing operator $u:X \to \ell_2$ satisfies $\gamma_1(u)\leq C\pi_1(u)$, that is, every such operator is 1-factorable with factorization norm controlled by $C$ times the 1-summing norm. In a separate geometric line of work, the notation $G_L(X)$ is used for a constant built from equilateral-triangle configurations on the unit sphere. Both notions are quantitative probes of Banach-space structure, but they belong to different frameworks and should not be conflated (Defant et al., 30 Sep 2025, Chen et al., 2021).

1. Terminological scope and basic definitions

The two notions can be separated as follows.

Notation	Definition	Setting
$\boldsymbol{gl}(X)$	smallest $C>0$ such that every 1-summing $u:X\to\ell_2$ satisfies $\gamma_1(u)\leq C\pi_1(u)$	local Banach space theory
$C>0$0	$C>0$1	metric geometry of the unit sphere

For $C>0$2, the operative objects are 1-summing and 1-factorable operators, projection constants, unconditional basis constants, and Sidon constants. For $C>0$3, the operative objects are triples $C>0$4, Gao’s constant $C>0$5, and convexity moduli. This suggests that the shared name reflects a terminological overlap rather than a common definition (Defant et al., 30 Sep 2025, Chen et al., 2021).

2. The operator-theoretic constant $C>0$6

In the local theory, $C>0$7 quantifies the extent to which 1-summing operators from $C>0$8 to $C>0$9 admit controlled 1-factorizations. In the Boolean-cube setting this is described as measuring the space’s “local unconditional structure.” A basic comparison principle stated for Banach spaces with a 1-unconditional basis is

$u:X \to \ell_2$0

Here $u:X \to \ell_2$1 is the unconditional basis constant and $u:X \to \ell_2$2 is the Sidon constant. Thus, in the presence of a 1-unconditional basis, the Gordon-Lewis constant collapses to more classical unconditionality parameters (Defant et al., 2023).

The significance of $u:X \to \ell_2$3 in the cited works is its role as a bridge constant. In both analytic polynomial spaces and Boolean cube function spaces, it is compared systematically with projection constants and unconditional basis constants. A plausible implication is that $u:X \to \ell_2$4 is especially useful when unconditionality is absent or badly behaved, but projection-theoretic or factorization-theoretic estimates remain available.

3. Analytic polynomial spaces and asymptotic estimates

For a finite-dimensional Banach space $u:X \to \ell_2$5 and a finite index set $u:X \to \ell_2$6, the space $u:X \to \ell_2$7 consists of analytic polynomials whose monomial coefficients vanish outside $u:X \to \ell_2$8, equipped with the supremum norm on the unit sphere of $u:X \to \ell_2$9. In this setting, the Gordon-Lewis constant is studied together with the projection constant $\gamma_1(u)\leq C\pi_1(u)$0 and the unconditional basis constant $\gamma_1(u)\leq C\pi_1(u)$1.

A central comparison theorem states that

$\gamma_1(u)\leq C\pi_1(u)$2

for index sets of degree at most $\gamma_1(u)\leq C\pi_1(u)$3. The accompanying interpretation given in the source is that the Gordon-Lewis constant is always bounded above by the unconditional basis constant, and up to an exponential-in-degree factor, the converse holds. A more refined estimate compares $\gamma_1(u)\leq C\pi_1(u)$4 to projection constants on reduced index sets: $\gamma_1(u)\leq C\pi_1(u)$5 and, for general degree up to $\gamma_1(u)\leq C\pi_1(u)$6,

$\gamma_1(u)\leq C\pi_1(u)$7

The reduced index set $\gamma_1(u)\leq C\pi_1(u)$8 is described as combinatorially encoding which degree-$\gamma_1(u)\leq C\pi_1(u)$9 monomials are reached by removing one “letter” from indices in $C$0 (Defant et al., 30 Sep 2025).

The asymptotic regime is controlled by projection estimates. From Kadets-Snobar’s theorem, the source derives

$C$1

and therefore, for a “reasonable” class of Banach sequence lattices,

$C$2

If $C$3 is a 2-convex Banach lattice and $C$4 contains all tetrahedral indices, then this upper bound is sharp: $C$5 The same framework is tied to Bohr’s phenomenon through the statement that Gordon-Lewis and unconditional constants directly govern the decay of the Bohr radius. For finite-dimensional Lorentz sequence spaces $C$6, the source gives the explicit bound

$C$7

Matching lower bounds are also stated in regimes with sufficient convexity or concavity, with precise logarithmic corrections where necessary (Defant et al., 30 Sep 2025).

4. Boolean cube function spaces

For a family $C$8, the space $C$9 consists of all real-valued functions on $G_L(X)$0 whose Walsh-Fourier support is contained in $G_L(X)$1, equipped with the supremum norm. The associated constants are the projection constant $G_L(X)$2, the Sidon constant $G_L(X)$3, and the Gordon-Lewis constant $G_L(X)$4.

A key comparison result is

$G_L(X)$5

and for degree $G_L(X)$6,

$G_L(X)$7

For homogeneous spaces,

$G_L(X)$8

The paper emphasizes that, although the Walsh basis is not unconditional in many $G_L(X)$9 when $\boldsymbol{gl}(X)$0, the Gordon-Lewis constant remains finite and closely tracks the projection constant. This is presented as a specific feature of Boolean cube spaces rather than a generic fact about finite-dimensional Banach spaces (Defant et al., 2023).

Several model cases are identified explicitly. For degree $\boldsymbol{gl}(X)$1,

$\boldsymbol{gl}(X)$2

so $\boldsymbol{gl}(X)$3. For the full space $\boldsymbol{gl}(X)$4, the space is isometric to $\boldsymbol{gl}(X)$5, so $\boldsymbol{gl}(X)$6. For sparse families consisting only of singletons or of cardinality $\boldsymbol{gl}(X)$7, the paper states

$\boldsymbol{gl}(X)$8

up to constants from Khintchine’s inequality. For arithmetic families, such as square-free support, projection constants and thus Gordon-Lewis constants behave as

$\boldsymbol{gl}(X)$9

The asymptotic analysis uses symmetrization, desymmetrization, tensor methods, central limit techniques, the moment method, Slutsky’s theorem, and Hermite polynomial asymptotics (Defant et al., 2023).

5. Lewis weights, $C>0$0-embeddings, and computational relevance

A separate but related computational direction enters through $C>0$1-Lewis weights. For a matrix $C>0$2, the Lewis weights $C>0$3 generalize leverage scores from $C>0$4 to $C>0$5, and are defined implicitly by

$C>0$6

with $C>0$7. The cited paper states that approximate Lewis weights are central to bounding the Gordon-Lewis constant, which controls the quality of $C>0$8-embeddings, oblivious subspace embeddings, and John ellipsoid approximation (Apers et al., 2024).

The algorithmic contribution emphasized there is a simple post-processing step that turns one-sided approximate $C>0$9-Lewis weights into two-sided approximations via one fixed-point iteration: $u:X\to\ell_2$0 The stated error bound is

$u:X\to\ell_2$1

and the main theorem asserts that, using

$u:X\to\ell_2$2

many

$u:X\to\ell_2$3

-approximate leverage score computations, one can compute a two-sided $u:X\to\ell_2$4-approximation of the $u:X\to\ell_2$5-Lewis weights. This suggests that quantitative control of Gordon-Lewis-type embedding parameters can be coupled to low-precision leverage-score primitives rather than only to high-accuracy fixed-point schemes.

6. The geometric constant $u:X\to\ell_2$6 attached to inscribed triangles

In a distinct geometric framework, $u:X\to\ell_2$7 is defined for a real Banach space $u:X\to\ell_2$8 by

$u:X\to\ell_2$9

The constraint requires $\gamma_1(u)\leq C\pi_1(u)$0, $\gamma_1(u)\leq C\pi_1(u)$1, and $\gamma_1(u)\leq C\pi_1(u)$2 to be unit vectors. For $\gamma_1(u)\leq C\pi_1(u)$3 with the Euclidean norm, this describes the vertices of an equilateral triangle inscribed in the unit sphere, and the vectors $\gamma_1(u)\leq C\pi_1(u)$4 and $\gamma_1(u)\leq C\pi_1(u)$5 correspond to associated linear configurations (Chen et al., 2021).

The basic bounds are

$\gamma_1(u)\leq C\pi_1(u)$6

In a real Hilbert space,

$\gamma_1(u)\leq C\pi_1(u)$7

because for $\gamma_1(u)\leq C\pi_1(u)$8, both $\gamma_1(u)\leq C\pi_1(u)$9 and $C>0$00 equal $C>0$01. For $C>0$02,

$C>0$03

realized, for example, by $C>0$04 and $C>0$05. The paper further states that if $C>0$06 and $C>0$07 is finite-dimensional, then $C>0$08 is not strictly convex, and that if $C>0$09 is not super-reflexive, then $C>0$10 (Chen et al., 2021).

The constant is linked to Gao’s constant

$C>0$11

which satisfies $C>0$12, equals $C>0$13 in inner product spaces, and has the implication $C>0$14 is uniformly non-square. The key comparison is

$C>0$15

together with the implication chain

$C>0$16

The paper also compares $C>0$17 with the James constant $C>0$18, the von Neumann-Jordan constant $C>0$19, and the Zbaganu constant $C>0$20. In particular, $C>0$21 is described as a higher degree, non-symmetrical analog of the James constant, using $C>0$22 rather than $C>0$23, and hence probing a different linear configuration associated to equilateral triangles on the sphere. Its relation to convexity is quantified by

$C>0$24

For inner product spaces, the paper notes both $C>0$25 and $C>0$26 (Chen et al., 2021).

The same work introduces the $C>0$27-parameterized family

$C>0$28

with bounds

$C>0$29

and comparison

$C>0$30

An associated product-type invariant is

$C>0$31

satisfying

$C>0$32

and $C>0$33. Within this geometric usage, the constant serves as a quantitative interface between equilateral configurations on the unit sphere and structural properties such as strict convexity, uniform non-squareness, and super-reflexivity (Chen et al., 2021).