Outer Perimeter in Diverse Domains

• Outer perimeter is a diverse concept describing various exterior-boundary functionals across geometric, combinatorial, and control-theoretic applications.
• Research quantifies outer perimeter through methods like visual-angle integrals, intrinsic norm-based line elements, and discrete neighbor counts tailored to each field.
• Studies in urban traffic, patrol, and defense leverage outer perimeter definitions to optimize boundary identification, flow control, and threat detection strategies.

“Outer perimeter” is not a single invariant across the arXiv literature. In different research programs it denotes, or is approximated by, the boundary of a protected traffic zone, the defended boundary of a pursuit–evasion game, the affine length of the boundary of a toric moment region, the number of exterior lattice cells adjacent to a combinatorial object, the visible portion of opaque boundaries under a stacking order, or the first-order outer volume growth of a set under Minkowski expansion. Several papers are explicit that ordinary polygonal perimeter, for example $\mathcal P(p_1,\dots,p_n)=|p_1p_2|+\cdots+|p_np_1|$, is a different notion and does not address hull, exterior-boundary, or outer-envelope questions (Taitler, 2024, Cristofaro-Gardiner et al., 30 Jun 2025, Mansour et al., 2020, Kiderlen et al., 4 Apr 2025, Khimshiashvili et al., 2020).

1. Terminological scope and domain dependence

In computational traffic and control settings, the relevant object is often a protected region together with the boundary through which inflow or intrusion occurs. In “perimeter identification,” the boundary is operationalized as the convex hull of selected intersections; in “perimeter defense,” it is the defended boundary of a compact region; and in heterogeneous perimeter control it is the boundary of a protected urban region together with its feeder links or entry intersections (Taitler, 2024, Bajaj et al., 2021, Li et al., 2024).

In geometric and symplectic settings, the same phrase points to different boundary functionals. The symplectic paper on generalized convex toric domains states that the phrase “outer perimeter” does not appear explicitly, and that the closest paper-defined notion is the $SL_2(\mathbb Z)$-invariant affine perimeter $\ell(\Omega)=\ell(\partial\Omega)$ of the moment region, or locally the affine length of an outer boundary arc $\ell(\partial^+\Omega')$ (Cristofaro-Gardiner et al., 30 Jun 2025). In convex geometry, perimeter may be recovered from exterior visual-angle data or replaced by the intrinsic “self-perimeter” of the unit ball of a normed space, again producing an outer-boundary quantity that is not simply Euclidean length (Bruna et al., 2024, Wolansky, 2 Apr 2026).

In discrete combinatorics, “outer perimeter” is often a neighbor-count rather than a contour length. For convex polyominoes, the outer-site perimeter is the number of cells in the complement with at least one common edge with a cell in the polyomino; for proper polycubes, perimeter is the number of adjacent cells that are empty; and for disk arrangements, visible perimeter is the total length of all pieces of disk boundaries visible under a stacking order (Mansour et al., 2020, Luther et al., 2017, Nivasch et al., 2012).

This diversity of definitions is not merely terminological. It indicates that “outer perimeter” is best understood as a family of exterior-boundary functionals whose exact meaning is fixed by the ambient structure: Euclidean, integral-affine, combinatorial, topological, or control-theoretic.

2. Convex-geometric, integral-geometric, and symplectic formulations

For a planar compact convex set $K\subset \mathbb R^2$, one classical exterior formulation reconstructs perimeter from visual-angle data. The paper on perimeter, area, and visual angle uses the Crofton-type identity

$$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$

and proves that Crofton’s formula is the unique universal formula relating visual angle, perimeter, and area. It also derives the asymptotic recovery formula

$$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$

so the first asymptotic term of the visual angle seen from far away determines the perimeter of the outer boundary (Bruna et al., 2024).

A different intrinsic construction appears in the theory of “self perimeter of convex sets.” In dimension $2$, if $B$ is a centrally symmetric convex set, the self-perimeter is

$$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$

where $SL_2(\mathbb Z)$0 is Euclidean arclength, $SL_2(\mathbb Z)$1 is the tangent vector, and $SL_2(\mathbb Z)$2 is the radial function. In higher dimension the paper defines perimeter recursively by integrating over $SL_2(\mathbb Z)$3 with an integrand involving the self-volume of the central section $SL_2(\mathbb Z)$4, and imposes the Euclidean normalization

$SL_2(\mathbb Z)$5

The construction is stated to be invariant under origin-preserving affine transformations and polar duality, and it agrees with the ordinary Euclidean value on Euclidean balls (Wolansky, 2 Apr 2026).

In symplectic geometry, the relevant outer-boundary functional is the affine perimeter of the moment region of a generalized convex toric domain. The paper defines $SL_2(\mathbb Z)$6 as the affine length of the boundary, not Euclidean perimeter, and emphasizes that smooth curved pieces or irrational-slope segments contribute zero. This perimeter appears in the subleading asymptotics of ECH and elementary ECH capacities through

$SL_2(\mathbb Z)$7

and yields a filling obstruction: $SL_2(\mathbb Z)$8 for any full filling by generalized convex toric domains. In this setting, outer perimeter is therefore a spectral invariant encoded in the asymptotic symplectic spectrum rather than an ordinary boundary length (Cristofaro-Gardiner et al., 30 Jun 2025).

Taken together, these works show that an outer boundary can be measured by visual-angle integrals, by an intrinsic norm-dependent line element, or by an integral-affine boundary length detected by symplectic capacities. The common theme is exterior interaction with the boundary, not a unique metric formula.

3. Measure-theoretic, topological, and statistical outer boundaries

The measure-theoretic notion closest to an anisotropic outer perimeter is the outer $SL_2(\mathbb Z)$9-Minkowski content. For compact $\ell(\Omega)=\ell(\partial\Omega)$0, the paper on lower-dimensional structuring elements defines

$\ell(\Omega)=\ell(\partial\Omega)$1

when the limit exists, and identifies it with the anisotropic perimeter

$\ell(\Omega)=\ell(\partial\Omega)$2

under suitable hypotheses. Its main novelty is that $\ell(\Omega)=\ell(\partial\Omega)$3 may be lower-dimensional. The paper proves that a weaker AFP-condition relative to $\ell(\Omega)=\ell(\partial\Omega)$4,

$\ell(\Omega)=\ell(\partial\Omega)$5

is sufficient for the existence of the $\ell(\Omega)=\ell(\partial\Omega)$6-Minkowski content of a rectifiable set, and it gives an example in $\ell(\Omega)=\ell(\partial\Omega)$7 that does not admit isotropic outer Minkowski content but admits outer $\ell(\Omega)=\ell(\partial\Omega)$8-Minkowski content for all two-dimensional disks $\ell(\Omega)=\ell(\partial\Omega)$9 (Kiderlen et al., 4 Apr 2025).

A related but container-based perspective appears in the approximation theory of finite-perimeter sets. If $\ell(\partial^+\Omega')$0 has finite perimeter with $\ell(\partial^+\Omega')$1, then any finite-perimeter set $\ell(\partial^+\Omega')$2 can be approximated by a smooth or polyhedral bounded set $\ell(\partial^+\Omega')$3 so that both the total perimeter $\ell(\partial^+\Omega')$4 and the relative perimeter $\ell(\partial^+\Omega')$5 are approximated, while $\ell(\partial^+\Omega')$6 has negligible intersection with $\ell(\partial^+\Omega')$7. In that case

$\ell(\partial^+\Omega')$8

so the perimeter outside the container becomes an unambiguous outer contribution (Carbotti et al., 19 Mar 2026).

A statistical version of outer-boundary size appears in graph-cut perimeter estimation. For $\ell(\partial^+\Omega')$9, with $K\subset \mathbb R^2$0, the estimator is the appropriately rescaled graph cut between sample points in $K\subset \mathbb R^2$1 and those in $K\subset \mathbb R^2$2. The target is the relative perimeter $K\subset \mathbb R^2$3, and the paper proves consistency when

$K\subset \mathbb R^2$4

It distinguishes the dense regime $K\subset \mathbb R^2$5 from the sparse regime $K\subset \mathbb R^2$6, and in dimensions $K\subset \mathbb R^2$7 derives asymptotic normality in the dense regime (Trillos et al., 2016).

At the purely topological end, multiset topology separates “exterior” from “boundary.” For an M-set $K\subset \mathbb R^2$8, the exterior is

$K\subset \mathbb R^2$9

while the boundary is

$$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$0

In that framework, boundary is the object closest to an outer perimeter, whereas exterior is the open outside region. The paper also proves that $$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$1 iff every nonempty open M-set contains a point of $$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$2, which is the multiset analogue of density (Mahanta et al., 2015).

These works collectively separate three questions that are often conflated: what lies outside a set, what part of the boundary is measured, and how that quantity is estimated from data. “Outer perimeter” can refer to any of these, depending on the formalism.

4. Discrete, combinatorial, and visibility-based outer perimeters

For convex polyominoes, the outer-site perimeter is defined combinatorially. If $$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$3 is a polyomino and $$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$4, then the outer-site perimeter $$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$5 is the number of cells in $$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$6 with at least one common edge with a cell in $$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$7. This differs from the ordinary perimeter $$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$8, which counts boundary edges rather than distinct exterior neighboring cells. The paper develops a column-by-column decomposition of the generating function

$$\int_{P\notin K}(w-\sin w)\,dP=\frac{L^{2}}{2}-\pi F$$9

and obtains an explicit generating function counting convex polyominoes by outer-site perimeter (Mansour et al., 2020).

For proper polycubes, perimeter again means an exterior neighbor count, but now in higher-dimensional lattice percolation. The paper states that the perimeter of a polycube is “the number of adjacent cells that are empty,” not the number of exposed faces. If a polycube is proper in $$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$0 dimensions, embedding it into $$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$1 adds $$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$2 extra adjacent empty cells, and the counts satisfy

$$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$3

In this context, outer perimeter is a lattice-neighbor statistic that encodes how much empty ambient space touches the cluster (Luther et al., 2017).

The notion changes again for opaque disk arrangements. Given a stacking order $$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$4 on a family $$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$5 of unit disks, the visible perimeter $$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$6 is the total length of all boundary arcs visible from below. It is not the outer perimeter of the planar union, because it depends on stacking order and can count arcs not belonging to the boundary of $$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$7. The paper proves that every $$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$8-dense $$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi,$$9-point set admits a stacking order with $2$0 visible perimeter, that for any stacking order on a $2$1-dense set one has $2$2 visible perimeter in the shrinking limit, and that the $2$3 grid is tight at order $2$4 (Nivasch et al., 2012).

A further discrete reinterpretation appears in tiling theory. For a polygonal tiling $2$5 of a region $2$6, the paper defines a strong perimeter tile to be a tile sharing a path of discrete length at least two with the boundary of $2$7. The perimeter of the tiling is then the number of strong perimeter tiles. This quantity is neither Euclidean boundary length nor mere boundary-tile count; it is a tilewise measure of substantial overlap with the outer boundary. In rhombic tilings of Elnitsky polygons, the paper proves sharp minimum and maximum strong perimeter values and classifies permutations whose tilings have minimal perimeter in two different senses (Tenner, 2018).

Across these combinatorial theories, outer perimeter is not a contour integral. It is a count of exterior sites, empty neighbors, visible arcs, or boundary-hugging tiles, each adapted to the ambient discrete model.

5. Perimeter as a defended boundary in patrol and pursuit–evasion

In patrol theory, perimeter is the defended boundary itself. In the continuous patrol model, patrollers depart from a base, traverse the entire perimeter exactly once, and return to the base. If attack duration is $2$8, each passing detects independently with probability $2$9, and the long-run dispatch rate is $B$0, then the optimal detection probability is

$B$1

The paper shows that visible and undercover patrols have the same optimal value when patrol departures are randomized appropriately, so visibility of the outer perimeter does not reduce the Stackelberg value in this model (Lin, 2019).

In one-dimensional perimeter defense, the defended set is the central interval $B$2, and the active perimeter is the pair of boundary points $B$3 and $B$4. Intruders are released at $B$5 or $B$6 and move inward with speed $B$7, while a defender vehicle moves with unit speed. The paper proves that no constant-competitive algorithm exists when

$B$8

that every algorithm is at best $B$9-competitive when

$$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$0

that the Sweeping algorithm is $$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$1-competitive when

$$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$2

and that Capture with Patience is $$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$3-competitive when

$$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$4

Here outer perimeter is the boundary to be reached by inward-moving threats, and the main issue is online scheduling under adversarial arrivals (Bajaj et al., 2021).

A more geometric defense model is the hemisphere perimeter-defense problem. A ground intruder tries to reach the base circle of a hemisphere of radius $$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$5, while an aerial defender moves on the hemisphere surface and seeks to arrive first at the same breach point. The practical paper does not rederive the differential-game solution but implements the strategy based on a uniquely defined optimal breaching point. It then quantifies discrepancy between first-order theory and realistic execution using two metrics: $$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$6, the terminal geodesic mismatch on the hemisphere in intruder-win settings, and $$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$7, the terminal defender–intruder separation in defender-win settings. The results show convergence toward first-order behavior as $$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$8 increases, and also show that a baseline strategy can slightly outperform the theoretically optimal strategy because online breaching-point computation consumes about $$P(B)= \int_{\partial B} \frac{ds}{r_B(\vec{\tau}(s))} \equiv \int_{\partial B}h_{B^*}\left(\vec{\tau}(s)\right)ds,$$9 inside a $SL_2(\mathbb Z)$00 loop (Lee et al., 2021).

The turret-defense model pushes the scheduling aspect further. In a planar conical environment, a central turret with finite range $SL_2(\mathbb Z)$01 and nonzero service time $SL_2(\mathbb Z)$02 must defend a concentric perimeter at radius $SL_2(\mathbb Z)$03 against radially incoming intruders. The offline problem is shown to be equivalent to a Travelling Repairperson Problem with Time Windows, and when $SL_2(\mathbb Z)$04 the reachability graph becomes acyclic, so the optimal policy is obtained by computing a longest path. In the online setting, the paper proves a lower bound of $SL_2(\mathbb Z)$05 on the best achievable competitive ratio in a general regime, and gives a $SL_2(\mathbb Z)$06-competitive Sweeping in Turret algorithm and a $SL_2(\mathbb Z)$07-competitive Dynamically Project and Capture algorithm in explicit parameter ranges (Bajaj et al., 2023).

These defense papers use “perimeter” in the literal operational sense of a boundary that must be guarded, crossed, or patrolled. Their main contribution is not a new geometry of boundary length, but the conversion of a defended outer boundary into routing, scheduling, and competitive-analysis problems.

6. Urban traffic perimeters: identification and heterogeneous control

In urban traffic management, “perimeter” denotes the boundary of a congested protected region whose inflow must be monitored or controlled. The sequential-decision formulation of perimeter identification defines the outer perimeter as the convex hull of a selected set of intersections. The search space is therefore not an arbitrary contour but a convex hull over chosen vertices. The paper models the task as a finite-horizon MDP $SL_2(\mathbb Z)$08, where states are selected intersection sets, actions add or remove intersections, and the reward is the change in a congestion-minus-penalty value over the hull. Conceptually, the objective is

$SL_2(\mathbb Z)$09

with $SL_2(\mathbb Z)$10 controlling whether the perimeter is conservative, balanced, or permissive. The demonstration on downtown Toronto uses Google Maps congestion heat maps and extracted intersections, and under $SL_2(\mathbb Z)$11, $SL_2(\mathbb Z)$12, and $SL_2(\mathbb Z)$13 the agent “was able to quickly identify the appropriate perimeters for different regularization terms.” The paper also stresses that the convex hull is not yet the final road-network perimeter; a deterministic post-processing step is required to obtain an operational traffic cordon (Taitler, 2024).

A complementary problem is how to meter flow across a known perimeter when congestion inside the protected region is spatially heterogeneous. The heterogeneous perimeter-control paper assumes a single protected region with feeder links $SL_2(\mathbb Z)$14 on its boundary and meters only incoming vehicles. Homogeneous control assigns the same permitted inflow $SL_2(\mathbb Z)$15 to all perimeter entries, whereas heterogeneous control redistributes the total allowed inflow $SL_2(\mathbb Z)$16 across feeder links using multi-hop downstream pressure. Starting from the 1-hop pressure

$SL_2(\mathbb Z)$17

the paper defines

$SL_2(\mathbb Z)$18

where $SL_2(\mathbb Z)$19 is the Markov transition matrix built from turning ratios. The second-stage allocation is a Softmax redistribution

$SL_2(\mathbb Z)$20

which preserves the total inflow and increases inflow at feeder links with larger downstream pressure. In the experiments, hops $SL_2(\mathbb Z)$21 were tested, performance improved with hop count and converged after about $SL_2(\mathbb Z)$22 hops, and the multi-hop approaches significantly outperformed homogeneous perimeter control in scenarios with high spatial heterogeneity while remaining robust to turning-ratio perturbations (Li et al., 2024).

These traffic papers make explicit a distinction that is often implicit elsewhere: an outer perimeter is not only a boundary to be delineated, but also a boundary across which flux must be distributed. Identification asks where the boundary is; perimeter control asks how much of the outside demand to admit through each boundary segment.

Outer perimeter is therefore best treated as a domain-dependent exterior-boundary concept rather than as a single definition. In convex geometry it may be reconstructed from visual angles; in symplectic geometry it may be an affine spectral invariant; in measure theory it may be an outer Minkowski content; in discrete models it may be a count of exterior neighbors or visible arcs; in patrol and defense it is the guarded boundary itself; and in urban traffic it is both a cordon to be inferred and an interface across which inflow is regulated.