Diametral ℓ1 Pressure

• Diametral ℓ1 Pressure is a quantitative invariant defined from the orbit hull of a minimal displacement point for fixed-point-free nonexpansive maps.
• It is computed as the infimum over finite-level functionals that assess how well signed ℓ1-normalized combinations avoid collapse.
• A positive pressure indicates robust ℓ1-type geometry that may force an isomorphic copy of ℓ1, thereby challenging the reflexivity of the ambient space.

Searching arXiv for the cited paper and topic.

Diametral $\ell_1$ pressure is a quantitative invariant of an orbit hull associated with a fixed-point-free nonexpansive map on a Banach space. It is introduced to measure how far a weakly compact orbit hull is from “collapsing” under signed $\ell_1$-normalized combinations, and it is formulated relative to a minimal displacement point $x_\infty$ and the normalized geometry of the orbit hull $C_1$. In the framework proposed in "On the Fixed Point Property in Reflexive Banach Spaces" (Alpay et al., 16 Sep 2025), positivity of this invariant signals robust $\ell_1$-type geometry inside the orbit hull; if such positivity were available uniformly in the fixed-point-free setting, it would force an isomorphic copy of $\ell_1$ in the ambient space and thereby contradict reflexivity.

1. Definition and basic construction

The invariant is defined from the orbit hull of a minimal displacement point. For a nonexpansive map $T:C\to C$, the paper considers

$$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$

with $\Delta>0$ in the fixed-point-free case under discussion. The point $x_\infty$ is a minimal displacement point, and the diameter $\ell_1$0 provides the normalization scale for all subsequent quantities (Alpay et al., 16 Sep 2025).

For each $\ell_1$1, the finite-level functional is

$\ell_1$2

The diametral $\ell_1$3-pressure is then

$\ell_1$4

This definition packages several operations into a single invariant. One first chooses $\ell_1$5 points in the orbit hull, then examines all signed $\ell_1$6-normalized linear combinations of the centered and diameter-normalized vectors, then takes the worst such combination, then optimizes over the $\ell_1$7-tuple, and finally requires a uniform lower bound over all $\ell_1$8. The resulting quantity is therefore explicitly anti-cancellation: $\ell_1$9 means that arbitrarily large finite selections can be made so that every signed $x_\infty$0-normalized combination remains uniformly away from $x_\infty$1.

A separation-sensitive variant is also defined: $x_\infty$2 and

$x_\infty$3

This version forces the selected points to be well separated and is intended to suppress cancellations produced by repeated or nearly repeated points.

2. Geometric meaning in nonexpansive dynamics

The construction is tied to the geometry of nonexpansive orbits. The paper presents the orbit hull $x_\infty$4 as the natural object once a nonexpansive map has no fixed point: $x_\infty$5 minimizes $x_\infty$6, the orbit $x_\infty$7 is trapped in a weakly compact convex set, and $x_\infty$8 records the spread of that orbit (Alpay et al., 16 Sep 2025).

The role of $x_\infty$9 is normalization. Diametral $C_1$0 pressure does not measure absolute size; it measures collapse relative to the size of the orbit hull. In this normalized form, the invariant distinguishes two geometric tendencies. If signed combinations can collapse toward $C_1$1, then the pressure is small or vanishes. If such collapse is obstructed uniformly across all finite levels, then the orbit hull behaves like an $C_1$2-sequence.

The sequence $C_1$3 is nonincreasing, so

$C_1$4

This monotonicity is structurally important. A positive value of $C_1$5 for one particular $C_1$6 gives only a finite-level certificate; it does not imply positive diametral $C_1$7 pressure. Larger tuples may introduce new cancellation patterns, and the global invariant is sensitive precisely to this possibility. The paper therefore treats $C_1$8 as a genuinely uniform invariant rather than a one-shot geometric statistic.

A plausible implication is that diametral $C_1$9 pressure isolates a failure mode in earlier fixed-point arguments: finite geometric spreading of an orbit hull is insufficient unless it persists uniformly under arbitrarily large selections.

3. Main theorem and the connection to $\ell_1$0 and the fixed point property

The central role of diametral $\ell_1$1 pressure is conditional. The paper states that if $\ell_1$2 is reflexive and for every fixed-point-free nonexpansive map $\ell_1$3 one has

$\ell_1$4

for the orbit hull $\ell_1$5 of a minimal displacement point $\ell_1$6, then $\ell_1$7 must have the fixed point property (FPP) (Alpay et al., 16 Sep 2025).

The mechanism proceeds through a lower $\ell_1$8 estimate. The key lemma asserts that if a sequence $\ell_1$9 satisfies

$\ell_1$0

then the map

$\ell_1$1

is an isomorphic embedding bounded below by $\ell_1$2. Consequently, $\ell_1$3 contains an isomorphic copy of $\ell_1$4.

Since a reflexive Banach space cannot contain an isomorphic copy of $\ell_1$5, positive pressure in every fixed-point-free configuration would contradict reflexivity. The argument may be summarized as

$\ell_1$6

The separated version is analogous: if $\ell_1$7 uniformly for all fixed-point-free nonexpansive maps, then $\ell_1$8 has the FPP.

The logical status of this result is explicit. Diametral $\ell_1$9 pressure is not presented as an unconditional characterization of the FPP. Rather, it is the quantitative invariant that would supply the missing step in an attempted proof: positivity would convert orbit-hull geometry into an $T:C\to C$0 obstruction to reflexivity.

4. Finite-level certificates, examples, and vanishing phenomena

A major theme of the paper is that finite-level positivity and global positivity are sharply different notions. If $T:C\to C$1 is finite with $T:C\to C$2, then

$T:C\to C$3

because repeated points allow exact cancellation. Hence

$T:C\to C$4

This shows that every finite orbit hull has zero diametral $T:C\to C$5 pressure, regardless of any positive finite-level values that may appear for smaller $T:C\to C$6 (Alpay et al., 16 Sep 2025).

The same vanishing mechanism appears in elementary symmetric configurations. For

$T:C\to C$7

choosing coefficients $T:C\to C$8 makes the normalized signed sum equal to $T:C\to C$9, so $$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$0. In an equilateral triangle in $$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$1, the paper gives

$$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$2

because the three vertices sum to zero under equal weights. These examples show that geometric spread alone does not prevent total signed cancellation.

By contrast, orthogonal configurations produce genuine finite-level lower bounds. For the orthonormal triple $$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$3 in $$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$4,

$$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$5

For an orthonormal $$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$6-tuple in a Hilbert space,

$$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$7

These are finite-dimensional witnesses that nearly orthogonal families resist $$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$8-collapse, but they remain finite-level statements.

The paper also gives a non-uniformly convex example in $$C_1 := \overline{\operatorname{conv}\{T^n x_\infty : n\ge 0\}}, \qquad \Delta := \operatorname{diam}(C_1),$$9: taking $\Delta>0$0 from the convex hull of the standard basis vectors leads to

$\Delta>0$1

because the $\Delta>0$2 mass can be spread among many coordinates so that the sup norm of the combination becomes arbitrarily small. This example clarifies that highly structured sets may still have zero pressure when the ambient norm permits substantial cancellation.

A recurring misconception addressed by these examples is that “well spread” finite geometry should force positive diametral pressure. The examples show that this is false: the infimum over all $\Delta>0$3 is the decisive feature.

5. Computational criteria and the weighted selection functional

The paper develops several finite-dimensional certification methods. For unit vectors $\Delta>0$4 in a Hilbert space, let

$\Delta>0$5

be the mutual coherence. If the Gram matrix $\Delta>0$6 has smallest eigenvalue $\Delta>0$7, then for $\Delta>0$8,

$\Delta>0$9

Using the coherence bound $x_\infty$0 and

$x_\infty$1

the paper obtains

$x_\infty$2

Thus low mutual coherence yields a certified positive lower bound at finite level (Alpay et al., 16 Sep 2025).

A dual functional certificate is also given. If $x_\infty$3 with $x_\infty$4 satisfies

$x_\infty$5

then

$x_\infty$6

Hence $x_\infty$7. In finite dimensions, the paper describes this as a convex optimization problem; in Euclidean or polyhedral settings, it can be checked numerically.

An unsigned companion invariant, the weighted selection functional, is defined by

$x_\infty$8

and

$x_\infty$9

Since nonnegative weights are more restrictive than arbitrary signed $\ell_1$00-normalized coefficients,

$\ell_1$01

Positivity of $\ell_1$02 is therefore a stronger certificate than positivity of diametral $\ell_1$03 pressure. The distinction is conceptual as well as technical: $\ell_1$04 measures resistance to collapse under convex combinations, whereas $\ell_1$05 measures resistance under signed combinations.

6. Relation to classical geometry, algorithmic aspects, and open issues

The paper contrasts diametral $\ell_1$06 pressure with normal structure, minimal displacement, and classical moduli of convexity and smoothness. Normal structure is a qualitative sufficient condition for fixed points via Kirk’s theorem; by contrast, $\ell_1$07 is described as a quantitative anti-collapse condition on orbit hulls. The invariant is also orbit-specific rather than a global parameter of the norm. Minimal displacement identifies how far the map is from having a fixed point, while diametral $\ell_1$08 pressure probes whether the associated orbit hull contains $\ell_1$09-type geometry (Alpay et al., 16 Sep 2025).

The computational side is unusually concrete. The paper includes an x86-64 assembly routine named coherence_phi_lower that reads a matrix of vectors, computes their Euclidean norms, computes all pairwise inner products, evaluates the maximum normalized absolute inner product $\ell_1$10, and returns both $\ell_1$11 and the lower bound for $\ell_1$12. The implemented estimate is

$\ell_1$13

This routine does not compute the full global invariant $\ell_1$14; it computes a certified lower estimate of finite-level diametral pressure in the Hilbert-space setting.

Several limitations are emphasized. Positive $\ell_1$15 at one level does not imply $\ell_1$16. Uniform convexity may help control finite $\ell_1$17, but it does not automatically force positive pressure for arbitrary orbit hulls. The paper therefore uses diametral $\ell_1$18 pressure to explain why existing approaches fail: they detect partial geometric spreading but do not rule out cancellation at arbitrarily large finite scales.

Open problems remain central to the concept. The paper is framed around the open question of whether every reflexive Banach space has the fixed point property. Diametral $\ell_1$19 pressure is proposed as a quantitative framework for that question, not as a completed resolution. This suggests that its lasting significance lies in converting a qualitative fixed-point problem into a family of orbit-hull anti-collapse estimates that can be studied analytically, geometrically, and computationally.