--- title: Perimetric Nonexpansive Mappings url: https://www.emergentmind.com/topics/perimetric-nonexpansive-mappings type: topic --- # Perimetric Nonexpansive Mappings 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” [2508.06821], 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 [2308.01003], higher-order total pairwise distance contractions [2402.02536], polygonal perimeter contractions [2410.20449], and perimetric contractions in generalized metric settings [2606.30954], while remaining distinct from the separate four-point “firm” metric framework developed for weak metric spaces [2110.13195]. ## 1. Definition and basic geometric idea The formal definition currently attached to the name *perimetric nonexpansive mapping* appears in [2508.06821]. 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 [2508.06821]. 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 [2508.06821]. This perimeter viewpoint has clear antecedents in the strict contractive literature. “Fixed point theorem for mappings contracting perimeters of triangles” [2308.01003] 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 \(\alpha=1\) of this earlier perimeter-contractive scheme, but the strict papers themselves do not use the term “perimetric nonexpansive” [2308.01003]. The same perimeter philosophy extends beyond triangles. “Periodic points of mappings contracting total pairwise distance” [2402.02536] defines \[ S(x_1,\dots,x_n)=\sum_{1\le i1=\|(1,0)-(0,0)\|_1. \] Thus perimetric nonexpansiveness properly enlarges the usual nonexpansive class [2508.06821]. The same example also shows that perimetric nonexpansiveness does not imply quasi-nonexpansiveness. Since \((0,0)\) is a fixed point, quasi-nonexpansiveness would require \[ \|Tx-(0,0)\|\le \|x-(0,0)\|. \] But for \(x=(1,0)\), \[ \|T(1,0)-(0,0)\|_1=2>1=\|(1,0)-(0,0)\|_1. \] So the map is perimetric nonexpansive but not quasi-nonexpansive [2508.06821]. The strict perimeter-contractive literature clarifies why this enlargement is natural. In [2308.01003], 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 [2508.06821]. | Class | Defining control | Inclusion facts in the cited literature | |---|---|---| | Nonexpansive | Pairwise distances | Included in perimetric nonexpansive [2508.06821] | | Quasi-nonexpansive | Distances to fixed points | Not implied by perimetric nonexpansive [2508.06821] | | Perimetric nonexpansive | Triangle perimeters | Strictly broader than nonexpansive [2508.06821] | | Perimetric contraction | Triangle perimeters with factor \(<1\) | Strict version studied earlier [2308.01003] | A separate nearby but nonidentical development is the notion of firm non-expansiveness in weak metric spaces [2110.13195]. There the key inequality is a four-point weighted metric relation involving \(\delta(x,y)\), \(\delta(x,Tx)\), \(\delta(y,Ty)\), \(\delta(x,Ty)\), and \(\delta(Tx,y)\). 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 [2110.13195]. ## 3. Elementary structure: continuity and fixed-point sets One of the first structural results in [2508.06821] is that every perimetric nonexpansive mapping on a normed linear space is continuous. The proof fixes \(x^*\in X\) and, when \(x^*\) is a limit point, chooses \(y\) close to \(x^*\) and applies the perimeter inequality to \(x,x^*,y\): \[ \|Tx-Tx^*\|+\|Tx^*-Ty\|+\|Ty-Tx\| \le \|x-x^*\|+\|x^*-y\|+\|y-x\|. \] Since the last two terms on the left are nonnegative, this yields \[ \|Tx-Tx^*\|\le 2(\|x-x^*\|+\|x^*-y\|), \] which gives continuity [2508.06821]. Continuity immediately implies that the fixed-point set \[ F(T):=\{x\in X:Tx=x\} \] is closed. If \(z_n\in F(T)\) and \(z_n\to z\), then continuity gives \[ Tz=\lim Tz_n=\lim z_n=z. \] Thus \(z\in F(T)\) [2508.06821]. This continuity phenomenon parallels earlier strict perimeter-contraction results. Both [2308.01003] and [2402.02536] prove continuity for maps contracting triangle perimeters or total pairwise distance, despite the fact that the assumptions involve triples or \(n\)-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 [2308.01003, 2402.02536]. At the same time, continuity alone is far from sufficient for fixed-point existence. The translation map on \(\mathbb{R}\), \[ Tx=x+1, \] is perimetric nonexpansive because \[ |Tx-Ty|+|Ty-Tz|+|Tz-Tx|=|x-y|+|y-z|+|z-x| \] for all \(x,y,z\), yet it has no fixed point [2508.06821]. 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 [2508.06821]. 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 \((X,\|\cdot\|)\) is a Banach space, \(K\) is a compact convex subset containing the null vector \(\theta\), and \(T:K\to K\) is perimetric nonexpansive with no periodic point of prime period \(2\), then \(T\) has a fixed point in \(K\) [2508.06821]. The proof regularizes \(T\) by \[ T_n(x)=\left(1-\frac1n\right)Tx. \] Each \(T_n\) is a strict perimeter-contraction with constant \(1-\frac1n<1\), so Petrov’s fixed-point theorem for perimeter-contractions applies once prime period-\(2\) points are excluded [2308.01003, 2508.06821]. A second compact-domain theorem in a general normed space assumes in addition that \((I-T)K\) is closed and that some orbit \(\{T^n x\}\) has a convergent subsequence. Under those assumptions, together with absence of prime period-\(2\) points, the limit point is a fixed point of \(T\) [2508.06821]. 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 \(K\) is a closed, bounded, convex subset of a Hilbert space \(H\) and \(T:K\to K\) is perimetric nonexpansive, then \(T\) has a fixed point [2508.06821]. No compactness of \(K\) and no exclusion of prime period-\(2\) points appear in the statement. The proof uses the regularization \[ U_n(x)=(1-s_n)x_0+s_nTx,\qquad s_n\in(0,1),\ s_n\to 1, \] with fixed \(x_0\in K\). Each \(U_n\) is a strict perimeter-contraction. Fixed points \(u_n\) of the \(U_n\) form an approximate fixed-point sequence for \(T\): \[ u_n=(1-s_n)x_0+s_nTu_n \quad\Longrightarrow\quad Tu_n-u_n=(1-s_n)(Tu_n-x_0), \] so \(\|Tu_n-u_n\|\to 0\). Weak compactness of closed bounded convex subsets of Hilbert spaces yields a weakly convergent subsequence \(u_{n_k}\rightharpoonup p\in K\), and Hilbert-space norm identities are then used to show \(Tp=p\) [2508.06821]. 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 [2508.06821]. ## 5. Periodic points, strict perimeter contractions, and approximation methods Periodic points play a central role in the background theory. The strict perimeter-contraction paper [2308.01003] 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 \(2\), and the number of fixed points is at most two. The proof iterates triangle perimeters \[ p_n=d(x_n,x_{n+1})+d(x_{n+1},x_{n+2})+d(x_{n+2},x_n), \] obtains \[ p_{n+1}\le \alpha p_n,\qquad \alpha<1, \] and deduces geometric decay and Cauchy convergence of the orbit [2308.01003]. The perimetric nonexpansive results of [2508.06821] 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 \(T\) by a family of strict contractions such as \(\left(1-\frac1n\right)T\) or \(U_n(x)=(1-s_n)x_0+s_nTx\). 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 \(\alpha<1\), while the perimetric nonexpansive theory operates at the threshold \(\alpha=1\), 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 [2308.01003, 2508.06821]. The broader perimeter-contraction literature reinforces this point. In the \(n\)-point setting of [2402.02536], strict total-pairwise-distance contraction yields periodic points of prime period at most \(n-1\). In polygonal perimeter contractions [2410.20449], fixed-point existence requires exclusion of periodic points of prime periods \(2,3,\dots,k-1\). In complete \(b\)-metric spaces [2606.30954], strict perimetric contractions lead to a dichotomy between a fixed-point regime and a pure \(2\)-cycle regime. None of these conclusions are established for the nonexpansive threshold, and the proofs depend essentially on strict constants \(<1\) [2402.02536, 2410.20449, 2606.30954]. ## 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 [2402.02536] and the polygonal perimeter framework of [2410.20449] both suggest higher-order analogues of perimetric nonexpansiveness obtained by replacing strict constants \(<1\) with nonincrease conditions. Those papers do not formulate such nonexpansive classes, but they provide the relevant functionals \[ S(x_1,\dots,x_n)=\sum_{i