---
title: Gordon-Lewis Constant in Banach Spaces
url: https://www.emergentmind.com/topics/gordon-lewis-constant
type: topic
---

# Gordon-Lewis Constant in Banach Spaces

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 [2509.26326] [2112.05922].

## 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 |
| \(G_L(X)\) | \(\sup\{\|x+y\|^2+\|2x-y\|^2:\|x\|=\|y\|=\|x-y\|=1\}\) | metric geometry of the unit sphere |

For \(\boldsymbol{gl}(X)\), the operative objects are 1-summing and 1-factorable operators, projection constants, unconditional basis constants, and Sidon constants. For \(G_L(X)\), the operative objects are triples \(x,y,x-y\in S_X\), Gao’s constant \(H(X)\), and convexity moduli. This suggests that the shared name reflects a terminological overlap rather than a common definition [2509.26326] [2112.05922].

## 2. The operator-theoretic constant \(\boldsymbol{gl}(X)\)

In the local theory, \(\boldsymbol{gl}(X)\) quantifies the extent to which 1-summing operators from \(X\) to \(\ell_2\) 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
\[
gl(X)=\chi(X)=\text{unconditional basis constant}=\mathrm{Sid}(X).
\]
Here \(\chi(X)\) is the unconditional basis constant and \(\mathrm{Sid}(X)\) is the Sidon constant. Thus, in the presence of a 1-unconditional basis, the Gordon-Lewis constant collapses to more classical unconditionality parameters [2302.00233].

The significance of \(\boldsymbol{gl}(X)\) 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 \(\boldsymbol{gl}(X)\) 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 \(X_n=(\mathbb{C}^n,\|\cdot\|)\) and a finite index set \(J\subset \mathbb{N}_0^n\), the space \(\mathcal{P}_J(X_n)\) consists of analytic polynomials whose monomial coefficients vanish outside \(J\), equipped with the supremum norm on the unit sphere of \(X_n\). In this setting, the Gordon-Lewis constant is studied together with the projection constant \(\boldsymbol{\lambda}(X)\) and the unconditional basis constant \(\boldsymbol{\chi}\).

A central comparison theorem states that
\[
\boldsymbol{gl}\big( \mathcal{P}_{J}(X_n)\big) \,\leq\,\boldsymbol{\chi}\big(\mathcal{P}_{J}(X_n) \big) \,\leq\, \boldsymbol{\chi}\big( \mathcal{P}_{J}(X_n) \big) \le 2^m\,\boldsymbol{gl}\big( \mathcal{P}_{J}(X_n) \big),
\]
for index sets of degree at most \(m\). 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 \(\boldsymbol{gl}\) to projection constants on reduced index sets:
\[
\boldsymbol{gl}\big( \mathcal{P}_{J}(X_n)\big) \leq e\cdot \|\mathbf{Q}_{\Lambda(m,n),J}\|\, \boldsymbol{\lambda}\big( \mathcal{P}_{J^\flat}(X_n)\big),
\]
and, for general degree up to \(m\),
\[
\boldsymbol{gl}\big( \mathcal{P}_{J}(X_n)\big) \le e (m+1)\, \max_{k}\|\mathbf{Q}_{\Lambda(k,n),J(k)}\|\cdot \max_{k} \boldsymbol{\lambda}\big( \mathcal{P}_{J(k)^\flat}(X_n)\big).
\]
The reduced index set \(J^\flat\) is described as combinatorially encoding which degree-\((m-1)\) monomials are reached by removing one “letter” from indices in \(J\) [2509.26326].

The asymptotic regime is controlled by projection estimates. From Kadets-Snobar’s theorem, the source derives
\[
\boldsymbol{\lambda}\big(\mathcal{P}_{J}(X_n)\big) \leq \sqrt{|J|}\leq \sqrt{m+1}\max_k \sqrt{|J(k)|},
\]
and therefore, for a “reasonable” class of Banach sequence lattices,
\[
\boldsymbol{gl}\big(\mathcal{P}_{J_m}(X_n)\big)  \prec_{C^m} \left( 1+\frac{n}{m} \right)^{m/2}.
\]
If \(X_n\) is a 2-convex Banach lattice and \(J_m\) contains all tetrahedral indices, then this upper bound is sharp:
\[
\boldsymbol{gl}\big(\mathcal{P}_{J_m}(X_n)\big)  \sim_{C^m}\left( 1+\frac{n}{m} \right)^{m/2}.
\]
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 \(\ell_{r,s}^n\), the source gives the explicit bound
\[
\boldsymbol{gl}\big(\mathcal{P}_{J_m}(\ell_{r,s}^n)\big) \leq C^m\left( \frac{n}{m} \right)^{m\cdot\min\left\{\frac{1}{2}, \frac{1}{r'}\right\} }.
\]
Matching lower bounds are also stated in regimes with sufficient convexity or concavity, with precise logarithmic corrections where necessary [2509.26326].

## 4. Boolean cube function spaces

For a family \(\mathcal{S}\subset 2^{[N]}\), the space \(\mathcal{B}_{\mathcal{S}}^N\) consists of all real-valued functions on \(\{-1,+1\}^N\) whose Walsh-Fourier support is contained in \(\mathcal{S}\), equipped with the supremum norm. The associated constants are the projection constant \(\lambda(\mathcal{B}_{\mathcal{S}}^N)\), the Sidon constant \(\mathrm{Sid}(\mathcal{B}_{\mathcal{S}}^N)\), and the Gordon-Lewis constant \(gl(\mathcal{B}_{\mathcal{S}}^N)\).

A key comparison result is
\[
gl(\mathcal{B}_\mathcal{S}^N) \leq \chi(\mathcal{B}_\mathcal{S}^N) \leq \mathrm{Sid}(\mathcal{B}_\mathcal{S}^N) \leq e^d\, gl(\mathcal{B}_\mathcal{S}^N),
\]
and for degree \(\leq d\),
\[
\mathrm{Sid}(\mathcal{B}_{\leq d}^N) \leq (2.69)^{2d} gl(\mathcal{B}_{\leq d}^N).
\]
For homogeneous spaces,
\[
gl(\mathcal{B}_d^N) \leq c_d \cdot \lambda(\mathcal{B}_d^N).
\]
The paper emphasizes that, although the Walsh basis is not unconditional in many \(\mathcal{B}_\mathcal{S}^N\) when \(d\geq 2\), 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 [2302.00233].

Several model cases are identified explicitly. For degree \(1\),
\[
\mathcal{B}_1^N \cong \ell_1^N,
\]
so \(gl=\chi=\mathrm{Sid}=\lambda \sim \sqrt{N}\). For the full space \(\mathcal{B}^N\), the space is isometric to \(\ell_\infty^{2^N}\), so \(gl=\chi=\lambda=1\). For sparse families consisting only of singletons or of cardinality \(m\), the paper states
\[
gl(\mathcal{B}_\mathcal{S}^N)\sim \sqrt{m},
\]
up to constants from Khintchine’s inequality. For arithmetic families, such as square-free support, projection constants and thus Gordon-Lewis constants behave as
\[
\lambda(\mathcal{B}_\mathcal{S}^N)\sim \sqrt{\frac{N}{\log\log N}}.
\]
The asymptotic analysis uses symmetrization, desymmetrization, tensor methods, central limit techniques, the moment method, Slutsky’s theorem, and Hermite polynomial asymptotics [2302.00233].

## 5. Lewis weights, \(\ell_p\)-embeddings, and computational relevance

A separate but related computational direction enters through \(\ell_p\)-Lewis weights. For a matrix \(A\in\mathbb{R}^{n\times d}\), the Lewis weights \(w_p(A)\) generalize leverage scores from \(p=2\) to \(p\geq 2\), and are defined implicitly by
\[
w^* = \big(A (A^\top W^{1 - 2/p} A)^{+} A^\top \big)_{\mathrm{diag}},
\]
with \(W=\mathrm{diag}(w^*)\). The cited paper states that approximate Lewis weights are central to bounding the Gordon-Lewis constant, which controls the quality of \(\ell_p\)-embeddings, oblivious subspace embeddings, and John ellipsoid approximation [2404.02881].

The algorithmic contribution emphasized there is a simple post-processing step that turns one-sided approximate \(\ell_p\)-Lewis weights into two-sided approximations via one fixed-point iteration:
\[
T_p(w)_i = w_i \cdot \left(\frac{\sigma_i(W^{1/2 - 1/p}A)}{w_i}\right)^{p/2}.
\]
The stated error bound is
\[
\| T_p(w) - w^* \|_1 \leq 3\left(\frac{p}{2} - 1\right) (1 + \varepsilon)^{\frac{p}{2} - 1} d,
\]
and the main theorem asserts that, using
\[
O(p d / \varepsilon)
\]
many
\[
O(\varepsilon / (p d))
\]
-approximate leverage score computations, one can compute a two-sided \(\varepsilon\)-approximation of the \(\ell_p\)-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 \(G_L(X)\) attached to inscribed triangles

In a distinct geometric framework, \(G_L(X)\) is defined for a real Banach space \(X\) by
\[
G_L(X) = \sup\{\|x+y\|^2 + \|2x-y\|^2 : \|x\| = \|y\| = \|x-y\| = 1\}.
\]
The constraint requires \(x\), \(y\), and \(x-y\) to be unit vectors. For \(X=\mathbb{R}^2\) with the Euclidean norm, this describes the vertices of an equilateral triangle inscribed in the unit sphere, and the vectors \(x+y\) and \(2x-y\) correspond to associated linear configurations [2112.05922].

The basic bounds are
\[
\frac{9}{2}\leq G_L(X)\leq 8.
\]
In a real Hilbert space,
\[
G_L(X)=6,
\]
because for \(x,y,x-y\in S_X\), both \(\|x+y\|^2\) and \(\|2x-y\|^2\) equal \(3\). For \(X=\ell_\infty\),
\[
G_L(\ell_\infty)=8,
\]
realized, for example, by \(x_0=(1,1,0,\ldots)\) and \(y_0=(1,0,0,\ldots)\). The paper further states that if \(G_L(X)=8\) and \(X\) is finite-dimensional, then \(X\) is not strictly convex, and that if \(X\) is not super-reflexive, then \(G_L(X)=8\) [2112.05922].

The constant is linked to Gao’s constant
\[
H(X)=\sup\left\{\min\{\|x+y\|,\|2x-y\|\}:x,y,x-y\in S_X\right\},
\]
which satisfies \(H(X)\leq 2\), equals \(\sqrt{3}\) in inner product spaces, and has the implication \(H(X)<2\Rightarrow X\) is uniformly non-square. The key comparison is
\[
H(X)^2 \leq \frac{1}{2}G_L(X),
\]
together with the implication chain
\[
G_L(X)<8 \;\Rightarrow\; H(X)<2 \;\Rightarrow\; X \text{ is uniformly non-square}.
\]
The paper also compares \(G_L(X)\) with the James constant \(J(X)\), the von Neumann-Jordan constant \(C_{\mathrm{NJ}}(X)\), and the Zbaganu constant \(C_Z(X)\). In particular, \(G_L(X)\) is described as a higher degree, non-symmetrical analog of the James constant, using \(\|2x-y\|\) rather than \(\|x-y\|\), and hence probing a different linear configuration associated to equilateral triangles on the sphere. Its relation to convexity is quantified by
\[
4(1-\delta_X(1))^2 \geq G_L(X)-4.
\]
For inner product spaces, the paper notes both \(G_L(X)=6\) and \(C_{\mathrm{NJ}}(X)=1\) [2112.05922].

The same work introduces the \(p\)-parameterized family
\[
G_L(X,p)=\sup\{\|x+y\|^p+\|2x-y\|^p:\|x\|=\|y\|=\|x-y\|=1\},\qquad p\in[1,\infty),
\]
with bounds
\[
2^{1-p}\cdot 3^p \leq G_L(X,p)\leq 2^{p+1},
\]
and comparison
\[
H(X)\leq 2^{-1/p}\,G_L(X,p)^{1/p}.
\]
An associated product-type invariant is
\[
C_L(X)=\sup\{\|x+y\|\cdot \|2x-y\|:\|x\|=\|y\|=\|x-y\|=1\},
\]
satisfying
\[
C_L(X)\leq \frac{1}{2}G_L(X),\qquad
C_L(X)\geq \frac{1}{2}(9-G_L(X)),
\]
and \(C_L(\ell_\infty)=4\). 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 [2112.05922].

Source: https://www.emergentmind.com/topics/gordon-lewis-constant