Perimetric Nonexpansive Mappings

• Perimetric nonexpansive mappings are nonlinear operators on normed linear spaces that satisfy a triangle perimeter control inequality.
• They generalize classical nonexpansiveness by allowing one distance to increase provided the overall triangle perimeter does not increase.
• Key fixed-point theorems in compact convex and Hilbert spaces are established using approximations via strict perimeter contractions.

Searching arXiv for papers on perimetric nonexpansive mappings and closely related perimeter-based mapping classes. Perimetric nonexpansive mappings are a class of nonlinear operators on normed linear spaces defined by a three-point metric inequality: instead of requiring that each pairwise distance be nonincreasing, they require only that the perimeter of every triangle not increase under the map. In the terminology introduced in “On a novel approach to nonexpansive mappings” (Banerjee et al., 9 Aug 2025), a mapping $T:X\to X$ on a normed linear space is perimetric nonexpansive if for all distinct $x,y,z\in X$,

$$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$

This perimeter-based condition places the theory between classical pairwise nonexpansiveness and broader multipoint metric dynamics. It is closely connected to earlier work on mappings contracting perimeters of triangles (Petrov, 2023), higher-order total pairwise distance contractions (Petrov, 2024), polygonal perimeter contractions (Zhou et al., 2024), and perimetric contractions in generalized metric settings (Abbas et al., 29 Jun 2026), while remaining distinct from the separate four-point “firm” metric framework developed for weak metric spaces (Gutiérrez et al., 2021).

1. Definition and basic geometric idea

The formal definition currently attached to the name perimetric nonexpansive mapping appears in (Banerjee et al., 9 Aug 2025). If $(X,\|\cdot\|)$ is a normed linear space, then $T:X\to X$ is perimetric nonexpansive when, for all distinct $x,y,z\in X$,

$$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$

The inequality compares the perimeter of the image triangle with the perimeter of the original triangle (Banerjee et al., 9 Aug 2025).

A useful interpretation is that the condition is intrinsically three-point rather than two-point. Classical nonexpansiveness requires

$\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$

so every edge is individually controlled. Perimetric nonexpansiveness controls only the sum of the three edge lengths. This allows one edge to increase provided the other two decrease enough to compensate. The defining mechanism is therefore global at the level of triangles rather than local at the level of single segments (Banerjee et al., 9 Aug 2025).

This perimeter viewpoint has clear antecedents in the strict contractive literature. “Fixed point theorem for mappings contracting perimeters of triangles” (Petrov, 2023) studies the strict version

$$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$

for pairwise distinct $x,y,z$. A plausible implication is that perimetric nonexpansiveness is the nonexpansive threshold $x,y,z\in X$0 of this earlier perimeter-contractive scheme, but the strict papers themselves do not use the term “perimetric nonexpansive” (Petrov, 2023).

The same perimeter philosophy extends beyond triangles. “Periodic points of mappings contracting total pairwise distance” (Petrov, 2024) defines

$x,y,z\in X$1

and studies maps satisfying

$x,y,z\in X$2

for pairwise distinct $x,y,z\in X$3-tuples. The case $x,y,z\in X$4 is exactly triangle perimeter contraction (Petrov, 2024). Likewise, “Perimetric Contraction on Polygons and Related Fixed Point Theorems” (Zhou et al., 2024) uses the polygonal perimeter

$x,y,z\in X$5

and imposes a strict factor $x,y,z\in X$6 on $x,y,z\in X$7-gons. These works establish a broader perimeter-based landscape in which perimetric nonexpansive mappings occupy the borderline nonincreasing regime rather than the strictly contractive one (Zhou et al., 2024).

2. Relation to classical nonexpansive and quasi-nonexpansive mappings

Every nonexpansive mapping is perimetric nonexpansive. This follows immediately by summing the three pairwise inequalities

$x,y,z\in X$8

to obtain the perimeter inequality (Banerjee et al., 9 Aug 2025).

The inclusion is strict. The principal example in (Banerjee et al., 9 Aug 2025) is defined on $x,y,z\in X$9 with the $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$0-norm and

$$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$1

with

$$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$2

The paper verifies that $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$3 is perimetric nonexpansive, but it is not nonexpansive because

$$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$4

Thus perimetric nonexpansiveness properly enlarges the usual nonexpansive class (Banerjee et al., 9 Aug 2025).

The same example also shows that perimetric nonexpansiveness does not imply quasi-nonexpansiveness. Since $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$5 is a fixed point, quasi-nonexpansiveness would require

$$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$6

But for $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$7,

$$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$8

So the map is perimetric nonexpansive but not quasi-nonexpansive (Banerjee et al., 9 Aug 2025).

The strict perimeter-contractive literature clarifies why this enlargement is natural. In (Petrov, 2023), every Banach contraction is shown to contract triangle perimeters, but there exist perimeter-contractive maps that are not ordinary contractions. A plausible implication is that the nonexpansive analogue should similarly be strictly broader than pairwise nonexpansiveness, which is confirmed directly in (Banerjee et al., 9 Aug 2025).

Class	Defining control	Inclusion facts in the cited literature
Nonexpansive	Pairwise distances	Included in perimetric nonexpansive (Banerjee et al., 9 Aug 2025)
Quasi-nonexpansive	Distances to fixed points	Not implied by perimetric nonexpansive (Banerjee et al., 9 Aug 2025)
Perimetric nonexpansive	Triangle perimeters	Strictly broader than nonexpansive (Banerjee et al., 9 Aug 2025)
Perimetric contraction	Triangle perimeters with factor $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert.$$9	Strict version studied earlier (Petrov, 2023)

A separate nearby but nonidentical development is the notion of firm non-expansiveness in weak metric spaces (Gutiérrez et al., 2021). There the key inequality is a four-point weighted metric relation involving $(X,\|\cdot\|)$0, $(X,\|\cdot\|)$1, $(X,\|\cdot\|)$2, $(X,\|\cdot\|)$3, and $(X,\|\cdot\|)$4. That theory is “perimetric” only in an informal sense; it is not the same notion as perimetric nonexpansiveness, and the paper does not use that terminology (Gutiérrez et al., 2021).

3. Elementary structure: continuity and fixed-point sets

One of the first structural results in (Banerjee et al., 9 Aug 2025) is that every perimetric nonexpansive mapping on a normed linear space is continuous. The proof fixes $(X,\|\cdot\|)$5 and, when $(X,\|\cdot\|)$6 is a limit point, chooses $(X,\|\cdot\|)$7 close to $(X,\|\cdot\|)$8 and applies the perimeter inequality to $(X,\|\cdot\|)$9: $T:X\to X$0 Since the last two terms on the left are nonnegative, this yields

$T:X\to X$1

which gives continuity (Banerjee et al., 9 Aug 2025).

Continuity immediately implies that the fixed-point set

$T:X\to X$2

is closed. If $T:X\to X$3 and $T:X\to X$4, then continuity gives

$T:X\to X$5

Thus $T:X\to X$6 (Banerjee et al., 9 Aug 2025).

This continuity phenomenon parallels earlier strict perimeter-contraction results. Both (Petrov, 2023) and (Petrov, 2024) prove continuity for maps contracting triangle perimeters or total pairwise distance, despite the fact that the assumptions involve triples or $T:X\to X$7-tuples of distinct points rather than two-point Lipschitz bounds. This suggests that perimeter control, even when weaker than pairwise nonexpansiveness, still imposes substantial regularity (Petrov, 2023, Petrov, 2024).

At the same time, continuity alone is far from sufficient for fixed-point existence. The translation map on $T:X\to X$8,

$T:X\to X$9

is perimetric nonexpansive because

$x,y,z\in X$0

for all $x,y,z\in X$1, yet it has no fixed point (Banerjee et al., 9 Aug 2025). This example marks a basic limitation of the theory: perimeter nonincrease does not by itself force any fixed-point property on arbitrary normed spaces.

4. Fixed-point theory on compact and Hilbertian domains

The main fixed-point results currently attached to perimetric nonexpansive mappings are those of (Banerjee et al., 9 Aug 2025). They are obtained by approximating a perimetric nonexpansive map by strict perimeter-contractions and then passing to the limit.

The first theorem concerns compact convex subsets of Banach spaces. If $x,y,z\in X$2 is a Banach space, $x,y,z\in X$3 is a compact convex subset containing the null vector $x,y,z\in X$4, and $x,y,z\in X$5 is perimetric nonexpansive with no periodic point of prime period $x,y,z\in X$6, then $x,y,z\in X$7 has a fixed point in $x,y,z\in X$8 (Banerjee et al., 9 Aug 2025). The proof regularizes $x,y,z\in X$9 by

$$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$0

Each $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$1 is a strict perimeter-contraction with constant $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$2, so Petrov’s fixed-point theorem for perimeter-contractions applies once prime period-$$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$3 points are excluded (Petrov, 2023, Banerjee et al., 9 Aug 2025).

A second compact-domain theorem in a general normed space assumes in addition that $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$4 is closed and that some orbit $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$5 has a convergent subsequence. Under those assumptions, together with absence of prime period-$$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$6 points, the limit point is a fixed point of $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$7 (Banerjee et al., 9 Aug 2025). This suggests that compactness alone is not the only route; asymptotic compactness information can substitute for full Banach-space completeness in the approximation argument.

The strongest result is the Hilbert-space theorem. If $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$8 is a closed, bounded, convex subset of a Hilbert space $$\lVert Tx-Ty \rVert+ \lVert Ty-Tz \rVert+ \lVert Tz-Tx \rVert \le \lVert x-y \rVert+ \lVert y-z \rVert+ \lVert z-x \rVert. \tag{2.1}$$9 and $\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$0 is perimetric nonexpansive, then $\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$1 has a fixed point (Banerjee et al., 9 Aug 2025). No compactness of $\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$2 and no exclusion of prime period-$\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$3 points appear in the statement. The proof uses the regularization

$\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$4

with fixed $\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$5. Each $\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$6 is a strict perimeter-contraction. Fixed points $\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$7 of the $\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$8 form an approximate fixed-point sequence for $\|Tx-Ty\|\le \|x-y\| \qquad \forall x,y,$9: $$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$0 so $$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$1. Weak compactness of closed bounded convex subsets of Hilbert spaces yields a weakly convergent subsequence $$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$2, and Hilbert-space norm identities are then used to show $$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$3 (Banerjee et al., 9 Aug 2025).

These results place perimetric nonexpansive mappings alongside classical fixed-point theory for nonexpansive maps on weakly compact convex subsets of Hilbert spaces. A plausible implication is that triangle-perimeter control retains enough structure to support fixed-point existence in Hilbert geometry even when pairwise nonexpansiveness is absent, but the present evidence is confined to the theorems explicitly proved in (Banerjee et al., 9 Aug 2025).

5. Periodic points, strict perimeter contractions, and approximation methods

Periodic points play a central role in the background theory. The strict perimeter-contraction paper (Petrov, 2023) proves that, on a complete metric space, a mapping contracting perimeters of triangles has a fixed point if and only if it does not possess periodic points of prime period $$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$4, and the number of fixed points is at most two. The proof iterates triangle perimeters

$$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$5

obtains

$$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$6

and deduces geometric decay and Cauchy convergence of the orbit (Petrov, 2023).

The perimetric nonexpansive results of (Banerjee et al., 9 Aug 2025) do not prove the same periodic-point characterization directly. Instead they use strict perimeter-contraction results as an approximation tool. The generic scheme is:

1. Replace $$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$7 by a family of strict contractions such as $$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$8 or $$d(Tx,Ty)+d(Ty,Tz)+d(Tx,Tz)\le \alpha\bigl(d(x,y)+d(y,z)+d(x,z)\bigr), \qquad \alpha\in[0,1),$$9.
2. Apply strict perimeter-contraction fixed-point theorems to the approximants.
3. Pass to the limit using compactness or weak compactness.

This method reveals an important structural difference between the contractive and nonexpansive regimes. The strict theory relies on factors $x,y,z$0, while the perimetric nonexpansive theory operates at the threshold $x,y,z$1, where geometric decay disappears. Consequently, the fixed-point arguments for perimetric nonexpansive maps require additional ambient structure, such as compactness or Hilbert weak compactness, rather than mere completeness (Petrov, 2023, Banerjee et al., 9 Aug 2025).

The broader perimeter-contraction literature reinforces this point. In the $x,y,z$2-point setting of (Petrov, 2024), strict total-pairwise-distance contraction yields periodic points of prime period at most $x,y,z$3. In polygonal perimeter contractions (Zhou et al., 2024), fixed-point existence requires exclusion of periodic points of prime periods $x,y,z$4. In complete $x,y,z$5-metric spaces (Abbas et al., 29 Jun 2026), strict perimetric contractions lead to a dichotomy between a fixed-point regime and a pure $x,y,z$6-cycle regime. None of these conclusions are established for the nonexpansive threshold, and the proofs depend essentially on strict constants $x,y,z$7 (Petrov, 2024, Zhou et al., 2024, Abbas et al., 29 Jun 2026).

6. Extensions, analogues, and conceptual boundaries

The present theory of perimetric nonexpansive mappings sits within a wider family of perimeter-based metric constructions.

The first natural extension concerns more than three points. The total pairwise distance framework of (Petrov, 2024) and the polygonal perimeter framework of (Zhou et al., 2024) both suggest higher-order analogues of perimetric nonexpansiveness obtained by replacing strict constants $x,y,z$8 with nonincrease conditions. Those papers do not formulate such nonexpansive classes, but they provide the relevant functionals

$x,y,z$9

and

$x,y,z\in X$00

This suggests a broader “polygonal nonexpansive” hierarchy, though such a theory remains external to the cited results (Petrov, 2024, Zhou et al., 2024).

A second extension concerns generalized metric spaces. “Perimetric Contractions and Their Iterates in Complete $x,y,z\in X$01-Metric Spaces” (Abbas et al., 29 Jun 2026) studies strict perimetric contractions in spaces where

$x,y,z\in X$02

The paper shows that sufficiently high iterates become graphic contractions under conditions like $x,y,z\in X$03, leading either to fixed points or to a unique isolated $x,y,z\in X$04-cycle. This is a strictly contractive theory, not a perimetric nonexpansive one, but it shows how sensitive perimeter dynamics become in generalized metric settings once strict decay is available (Abbas et al., 29 Jun 2026).

A third direction concerns best proximity points. “Best Proximity Point Results for Perimetric Contractions” (Garai et al., 31 Jan 2026) introduces perimetric proximal contractions of the first and second kind, again with strict factor $x,y,z\in X$05, and proves that best proximity points are not necessarily unique but that at most two may exist. A plausible implication is that a future theory of perimetric nonexpansive mappings for non-self maps would likely be formulated by relaxing those same three-point proximal inequalities from $x,y,z\in X$06 to $x,y,z\in X$07, but the paper itself does not carry out that extension (Garai et al., 31 Jan 2026).

The main conceptual boundary is terminological and structural. The four-point “firm” inequality of (Gutiérrez et al., 2021) is sometimes adjacent in spirit because it is fully metric and does not rely on convex combinations, but it is not a triangle-perimeter condition and should not be conflated with perimetric nonexpansiveness. Likewise, generic-behavior papers on ordinary nonexpansive mappings in Banach or CAT$x,y,z\in X$08 settings study pairwise Lipschitz phenomena, porosity, and Rakotch contractivity rather than perimeter-based classes (Dymond, 2021, Bargetz et al., 2020).

In this sense, perimetric nonexpansive mappings represent a distinct three-point branch of nonexpansive-type operator theory. Their defining feature is the replacement of edgewise metric control by control of triangle perimeters; their known fixed-point theory is strongest on compact convex subsets of Banach spaces and on closed bounded convex subsets of Hilbert spaces; and their current development remains tightly linked to the earlier strict perimeter-contraction literature from which many of their methods are derived (Banerjee et al., 9 Aug 2025, Petrov, 2023).