Increasing Chord Property in Geometric Paths

• The increasing chord property is a metric condition for directed curves ensuring that for any four ordered points, the outer chord is at least as long as the inner chord, reflecting a ‘no backtracking’ behavior.
• It underpins key results in both planar and higher-dimensional geometry, including sharp bounds like 2π/3 in the plane and exponential bounds in ℝᵈ, while also having a normed-plane generalization.
• The property drives algorithmic advancements in testing polygonal chains, optimizing shortest paths in simple polygons, and designing increasing-chord graphs for efficient routing.

The increasing chord property is a global metric monotonicity condition for directed curves and paths: if $a,b,c,d$ appear in this order on the curve, then the outer chord is not shorter than the inner chord, i.e. $|bc|\le |ad|$ in Euclidean space or $\|b-c\|\le \|a-d\|$ in a normed plane. In current research it is studied for piecewise smooth curves, polygonal chains, shortest paths in simple polygons, and geometric graph drawings. A central theme is that the property is equivalent to bidirectional self-approaching behavior and therefore imposes a strong “no backtracking” constraint, with sharp planar length bounds, explicit higher-dimensional bounds, and nontrivial algorithmic and complexity consequences (Dumitrescu et al., 1 Sep 2025, Lángi et al., 2 Sep 2025, Hagedoorn et al., 2022).

1. Definition and equivalent formulations

Let $\gamma:[0,1]\to \mathbb{R}^d$ be a curve with endpoints $s=\gamma(0)$ and $t=\gamma(1)$. The increasing chord property means that for every four points $a,b,c,d\in \gamma$ lying in this order along $\gamma$, one has

$|bc| \le |ad|.$

The same formulation is used in a normed plane, replacing Euclidean distance by the norm distance $\|\cdot\|_M$ (Dumitrescu et al., 1 Sep 2025, Lángi et al., 2 Sep 2025).

A basic equivalence relates increasing chords to self-approaching curves. A directed curve is self-approaching if $|bc|\le |ad|$0 whenever $|bc|\le |ad|$1 lie in this order on the curve. The higher-dimensional Euclidean theory recalls the well-known equivalence that a path has increasing chords if and only if both the path and its reversal are self-approaching; the polygonal-path literature in simple polygons states the same equivalence and notes that every subpath is then radially monotone in both directions (Dumitrescu et al., 1 Sep 2025, Hagedoorn et al., 2022).

For piecewise smooth Euclidean curves there is also a normal or halfspace characterization. At a point $|bc|\le |ad|$2, a normal is any hyperplane in the double wedge determined by the one-sided tangents; a positive halfspace $|bc|\le |ad|$3 contains a neighborhood of the forward subcurve and a negative halfspace $|bc|\le |ad|$4 contains a neighborhood of the backward subcurve. Then an $|bc|\le |ad|$5-$|bc|\le |ad|$6 curve has increasing chords if and only if any normal at any point $|bc|\le |ad|$7 does not intersect the open subcurves $|bc|\le |ad|$8 and $|bc|\le |ad|$9. Equivalently,

$\|b-c\|\le \|a-d\|$0

For polygonal chains, it is sufficient to check extremal normals at vertices (Dumitrescu et al., 1 Sep 2025). In the planar shortest-path setting this is expressed in the same way with normal lines and positive and negative half-planes (Hagedoorn et al., 2022).

Several representative examples are explicit. A straight line segment trivially has increasing chords. In the plane, an orthogonal staircase that always moves in the positive $\|b-c\|\le \|a-d\|$1- and positive $\|b-c\|\le \|a-d\|$2-direction has the property by the halfspace characterization, and its length is at most $\|b-c\|\le \|a-d\|$3, with equality attainable when the total horizontal and vertical displacements are equal. The literature also uses non-examples in which a witness normal line intersects both earlier and later portions of the curve on the wrong sides, showing how the halfspace condition fails (Dumitrescu et al., 1 Sep 2025).

2. Planar Euclidean theory

The planar case was settled through a sequence of results summarized in the higher-dimensional paper. Binmore asked whether there is a universal constant $\|b-c\|\le \|a-d\|$4 such that any plane curve with increasing chords from $\|b-c\|\le \|a-d\|$5 to $\|b-c\|\le \|a-d\|$6 has length at most $\|b-c\|\le \|a-d\|$7. Larman and McMullen proved one can take $\|b-c\|\le \|a-d\|$8, and Rote later proved that the optimal constant is

$\|b-c\|\le \|a-d\|$9

with this value best possible. Thus every planar curve with increasing chords satisfies

$\gamma:[0,1]\to \mathbb{R}^d$0

and the bound is tight (Dumitrescu et al., 1 Sep 2025).

The extremal examples are arcs consisting of two consecutive sides of a Reuleaux triangle. A Reuleaux triangle is obtained as the intersection of three unit disks centered at the vertices of an equilateral triangle, and taking the union of two adjacent circular arcs between opposite points $\gamma:[0,1]\to \mathbb{R}^d$1 yields an increasing-chord arc whose length achieves the ratio $\gamma:[0,1]\to \mathbb{R}^d$2 (Dumitrescu et al., 1 Sep 2025). The normed-plane paper presents the same Euclidean result as Rote’s theorem and describes the proof strategy as a reduction to canonical curves built from segments and involute pieces, followed by a rearrangement into a convex curve inside the intersection of two unit circles, whose perimeter gives the bound (Lángi et al., 2 Sep 2025).

The planar shortest-path problem inside a simple polygon adds an optimization layer. For two points $\gamma:[0,1]\to \mathbb{R}^d$3 in a simple polygon $\gamma:[0,1]\to \mathbb{R}^d$4, the shortest $\gamma:[0,1]\to \mathbb{R}^d$5-$\gamma:[0,1]\to \mathbb{R}^d$6 path with increasing chords is unique (Hagedoorn et al., 2022). The proof uses a structural lemma: if $\gamma:[0,1]\to \mathbb{R}^d$7 and $\gamma:[0,1]\to \mathbb{R}^d$8 are two distinct increasing-chord $\gamma:[0,1]\to \mathbb{R}^d$9-$s=\gamma(0)$0 paths in $s=\gamma(0)$1, then the geodesic in the region between them also has increasing chords. If two distinct shortest increasing-chord paths existed, that geodesic would be shorter, contradicting minimality (Hagedoorn et al., 2022).

The same work characterizes the shortest increasing-chord path as the geodesic in

$s=\gamma(0)$2

where $s=\gamma(0)$3 and $s=\gamma(0)$4 are the dead regions for self-approaching paths to $s=\gamma(0)$5 and $s=\gamma(0)$6, respectively. The resulting path may include arcs of dead-region boundaries rather than only straight segments. The boundaries are defined by transcendental equations and “likely cannot be solved or evaluated analytically,” so the method is a conceptual reduction unless one assumes exact solutions of those equations or accepts numerical approximation (Hagedoorn et al., 2022).

3. Higher-dimensional Euclidean geometry

The first explicit higher-dimensional length bounds were obtained for curves with increasing chords in $s=\gamma(0)$7, $s=\gamma(0)$8. For $s=\gamma(0)$9, the principal theorem states that for every $t=\gamma(1)$0,

$t=\gamma(1)$1

and, by choosing a suitable $t=\gamma(1)$2,

$t=\gamma(1)$3

This is the first bound in higher dimensions, and the paper notes that the first formula actually holds for arbitrary continuous curves with increasing chords, not only for piecewise smooth ones (Dumitrescu et al., 1 Sep 2025).

A key structural fact is monotonicity in the endpoint direction. If $t=\gamma(1)$4 has increasing chords, then the scalar function

$t=\gamma(1)$5

is monotone nondecreasing. This allows the curve to be confined using several monotone directions simultaneously. If a curve is monotone in $t=\gamma(1)$6 linearly independent directions $t=\gamma(1)$7, then it lies in the parallelotope

$t=\gamma(1)$8

and its length is bounded by the sum of the lengths of any $t=\gamma(1)$9 successive edges of $a,b,c,d\in \gamma$0 leading from $a,b,c,d\in \gamma$1 to $a,b,c,d\in \gamma$2. The proof of the higher-dimensional bound then constructs additional monotone directions inductively and controls the corresponding edge lengths by determinant estimates (Dumitrescu et al., 1 Sep 2025).

The resulting upper bound is exponential in $a,b,c,d\in \gamma$3. The authors explicitly state that it is “far from the truth” and pose as an open problem whether one can obtain a bound polynomial in $a,b,c,d\in \gamma$4 (Dumitrescu et al., 1 Sep 2025). No matching lower bound of similar order is known; even in $a,b,c,d\in \gamma$5, the best known examples have length about $a,b,c,d\in \gamma$6, from a construction of Rote (Dumitrescu et al., 1 Sep 2025).

A long-standing conjecture of Croft, Falconer, and Guy suggested that in $a,b,c,d\in \gamma$7 the longest increasing-chord curve should be an arc consisting of $a,b,c,d\in \gamma$8 consecutive sides of a Reuleaux simplex. This is false for every fixed $a,b,c,d\in \gamma$9. For a Reuleaux unit simplex in $\gamma$0, the midpoints of two disjoint edges are at distance

$\gamma$1

yielding counterexamples to the conjecture. In $\gamma$2, Rote had already refuted the conjecture by exhibiting a curve of length about $\gamma$3 with increasing chords (Dumitrescu et al., 1 Sep 2025). This makes extremal behavior in higher dimensions a genuinely open geometric problem rather than a straightforward extension of the planar Reuleaux-triangle picture.

4. Normed-plane generalization

A strictly convex normed plane replaces Euclidean disks by translates of an origin-symmetric convex disk $\gamma$4, with distance measured by $\gamma$5. In this setting the increasing chord property is defined in the same four-point way: $\gamma$6 for any $\gamma$7 appearing in this order on the curve (Lángi et al., 2 Sep 2025).

The main theorem for strictly convex normed planes is a direct analogue of Rote’s Euclidean bound. If $\gamma$8 and $\gamma$9 is a continuous curve with increasing chords from $|bc| \le |ad|.$0 to $|bc| \le |ad|.$1, then

$|bc| \le |ad|.$2

Thus the optimal norm-dependent upper bound is one-half of the perimeter of the lens formed by the two unit disks centered at the endpoints. In the Euclidean plane this recovers $|bc| \le |ad|.$3, since the relevant lens has perimeter $|bc| \le |ad|.$4 (Lángi et al., 2 Sep 2025).

Unlike the Euclidean case, the quantity

$|bc| \le |ad|.$5

depends on the direction of the segment $|bc| \le |ad|.$6. The paper therefore studies extremal values over all norms and all unit directions. It proves

$|bc| \le |ad|.$7

Parallelogram norms and affinely regular hexagon norms realize the minimum value $|bc| \le |ad|.$8 in the stated senses, the Euclidean norm yields the lower bound $|bc| \le |ad|.$9 for $\|\cdot\|_M$0, and parallelogram norms realize the maximum value $\|\cdot\|_M$1 (Lángi et al., 2 Sep 2025).

The proof relies on involutes in normed planes. For a convex disk $\|\cdot\|_M$2 and a point $\|\cdot\|_M$3, the involute $\|\cdot\|_M$4 is defined geometrically using oriented support lines and signed $\|\cdot\|_M$5-distance. The crucial properties are that $\|\cdot\|_M$6 is convex on every interval $\|\cdot\|_M$7, strictly convex there under strict convexity of the norm, and that $\|\cdot\|_M$8 has the increasing chord property with respect to the whole disk $\|\cdot\|_M$9. Involutes serve as canonical building blocks: arbitrary increasing-chord curves can be replaced, without increasing length, by concatenations of involute arcs and straight segments, then convexified and confined to the endpoint lens (Lángi et al., 2 Sep 2025).

The normed-plane paper also emphasizes a higher-dimensional contrast. In the $|bc|\le |ad|$00-dimensional $|bc|\le |ad|$01-norm, there exist increasing-chord curves $|bc|\le |ad|$02 with endpoints unit distance apart and

$|bc|\le |ad|$03

which grows exponentially with dimension (Lángi et al., 2 Sep 2025). This suggests that the geometric behavior of increasing chords depends strongly on the ambient norm, even when the planar Euclidean theory is sharp and rigid.

5. Algorithms and optimization problems

For polygonal chains in $|bc|\le |ad|$04, the halfspace characterization leads directly to testing algorithms. Given a polygonal chain $|bc|\le |ad|$05, a naive four-point test over all index quadruples is $|bc|\le |ad|$06, but the halfspace criterion reduces the task to emptiness queries for extremal halfspaces at vertices. A simple algorithm runs two loops: one checks negative halfspaces of segments $|bc|\le |ad|$07 against the remaining suffix, and the other checks positive halfspaces of segments $|bc|\le |ad|$08 against the remaining prefix. Scanning makes each query $|bc|\le |ad|$09, so the overall running time is $|bc|\le |ad|$10 (Dumitrescu et al., 1 Sep 2025).

The main algorithmic improvement is the first subquadratic higher-dimensional algorithm. For $|bc|\le |ad|$11, with $|bc|\le |ad|$12, one can determine by a randomized algorithm whether a polygonal chain satisfies the increasing chord property in

$|bc|\le |ad|$13

expected time. The method uses halfspace emptiness data structures based on optimal partition trees together with Bentley-style search decomposability and a block partitioning trick that balances brute-force work within the current block against static range-search queries on later blocks (Dumitrescu et al., 1 Sep 2025).

The polygonal shortest-path problem in a simple polygon is algorithmically different. The shortest $|bc|\le |ad|$14-$|bc|\le |ad|$15 path with increasing chords is unique, and it is the geodesic in the polygon after removing the two dead regions $|bc|\le |ad|$16 and $|bc|\le |ad|$17 associated with self-approaching feasibility to the two endpoints (Hagedoorn et al., 2022). This gives a conceptual algorithm: compute the two dead regions, form

$|bc|\le |ad|$18

and compute the geodesic there. The main bottleneck is that the dead-region boundaries are defined by transcendental equations, so the paper does not provide a standard finite-arithmetic complexity bound (Hagedoorn et al., 2022).

Open algorithmic questions remain explicit in both settings. The higher-dimensional testing paper asks whether one can improve the expected running time to

$|bc|\le |ad|$19

by exploiting the semi-dynamic, fixed-order deletion pattern, and also asks whether subquadratic algorithms are possible in low dimensions $|bc|\le |ad|$20 (Dumitrescu et al., 1 Sep 2025). The shortest-path paper asks whether there is an efficient algorithm for computing the shortest increasing-chord path in a polygon with holes and notes that the problem may be NP-hard (Hagedoorn et al., 2022).

6. Graph drawings, routing, and complexity

In geometric graph theory, an increasing-chord graph is a geometric graph in which for every unordered pair of vertices $|bc|\le |ad|$21, there exists a path between them that is self-approaching in both directions (Dehkordi et al., 2014). This makes increasing-chord graphs a strict strengthening of self-approaching graphs, and the literature stresses their routing relevance: self-approaching graphs are greedy, and increasing-chord graphs inherit the strong detour bound $|bc|\le |ad|$22 established for increasing-chord curves in the plane (Dehkordi et al., 2014, Nöllenburg et al., 2014).

Several existence theorems are known for point sets in the plane. Every Gabriel triangulation is an increasing-chord graph, proved via $|bc|\le |ad|$23-paths whose edges all lie in a fixed $|bc|\le |ad|$24 directional cone and are therefore self-approaching in both directions (Dehkordi et al., 2014). Using a result of Bern, Eppstein, and Gilbert, every planar point set with $|bc|\le |ad|$25 points admits an increasing-chord planar graph with $|bc|\le |ad|$26 Steiner points, constructible in $|bc|\le |ad|$27 time. For every convex point set with $|bc|\le |ad|$28 points, there also exists an increasing-chord graph on the original vertices with $|bc|\le |ad|$29 edges and no Steiner points, although that construction need not be planar (Dehkordi et al., 2014).

Recognition and spanning problems are substantially harder. Given a straight-line drawing $|bc|\le |ad|$30 and a root $|bc|\le |ad|$31, deciding whether $|bc|\le |ad|$32 contains an increasing-chord spanning tree rooted at $|bc|\le |ad|$33 is NP-complete (Bahoo et al., 2017). The same paper conjectures that deciding whether there exists an increasing-chord path between two given vertices is also NP-complete, and gives a reduction from 3-SAT, but the coordinate-size issue remains unresolved, so the complexity of the path problem is still open (Bahoo et al., 2017).

For planar graph classes, the Euclidean and hyperbolic pictures differ sharply. In the Euclidean plane, triangulations admit increasing-chord drawings, and planar 3-trees admit planar increasing-chord drawings via $|bc|\le |ad|$34-Schnyder drawings (Nöllenburg et al., 2014). The same work proves that strongly monotone, and hence increasing-chord, drawings of trees and binary cactuses may require exponential resolution, and it gives a binary cactus that does not admit a self-approaching drawing (Nöllenburg et al., 2014). By contrast, in the hyperbolic plane every 3-connected planar graph admits an increasing-chord drawing, and trees admit such drawings exactly when they are binary or subdivisions of $|bc|\le |ad|$35 (Nöllenburg et al., 2014).

A frequent source of confusion is terminology. In the curve and graph-drawing literature, “increasing chord property” refers to the metric four-point condition on paths and curves. A separate homological-combinatorial literature describes classical graph chordality as “the increasing chord property in dimension 1,” because every cycle of length at least $|bc|\le |ad|$36 is forced to acquire a chord; that framework extends to higher-dimensional simplicial complexes through resolution and decomposition chordality, Leray properties, and regularity (Adiprasito et al., 2015). This suggests a formal analogy, but it is a distinct theory from the metric geometry of increasing-chord curves and paths.