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Equally spaced points on a metric circle: stationary energy and the equality case

Example

Assume the Axiom of Choice for the cited short-loop criterion.

Let $0 < ℓ < 2 π$ and let $S_{ℓ}^{1} = R / ℓ Z$ be the circle of circumference $ℓ$ with its metric $d_{ℓ}$ (Comparison triangles, the CAT(0) and CAT(1) inequalities, local CAT, local geodesics and round circles, Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences (vi)). Fix $n \geq 5$ , choose $0 < ε < ℓ (1 / 4 - 1 / n)$ , and let $x$ be the cyclic $n$ -tuple of equally spaced points $0, ℓ / n, \dots, (n - 1) ℓ / n$ ; take the uniform radius $l = ℓ / 4 - ε$ of Uniform local radii, cyclic small-mesh polygons, mesh, length, energy, the midpoint operation and the zero-limit basin so that $mesh (x) = ℓ / n < l$ , and fix $h$ with $ℓ / n \leq h < l$ . Thus $x \in P_{h} (n)$ . Then:

(i) $mesh (x) = ℓ / n$ , $L (x) = ℓ$ , $E (x) = ℓ^{2} / n$ , and $f (x)$ is the equally spaced tuple shifted by $ℓ / (2 n)$ ; hence $L (f^{k} x) = ℓ$ and $E (f^{k} x) = ℓ^{2} / n$ for every $k$ , and $x \notin C_{h}^{0} (n)$ (Open ball, closed ball and sphere in a metric space).

(ii) Every consecutive triple of $x$ is straight, so $x$ realizes the equality case of Existence of the uniform radius, continuity of the midpoint operation, the energy drop, its equality case, and convergence of zero-limit polygons (iii): it is the list of $n$ equally spaced points of the closed local geodesic $S_{ℓ}^{1}$ . This is the exclusion of The uniform energy decrement on the basin, bounded iteration, and the closedness of the basin inside the short polygon space (iii): the basin can contain no such tuple.

(iii) $S_{ℓ}^{1}$ is compact and locally CAT(1) but not CAT(1) (Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences (vi)), and it contains the isometrically embedded circle $S_{ℓ}^{1}$ of length $ℓ < 2 π$ . The loop is a short nonshrinkable loop, in agreement with Polygon transfer, the basin as the shrinkable class, and the short-loop criterion (iii) with $m = ℓ$ ; in particular the tuple with stationary length and energy in (i) is not contradictory, because the quantitative decrement is stated on the basin only.

(iv) For $ℓ = 2 π$ the same computation gives a rotating tuple with stationary length and energy on the CAT(1) circle $S_{2 π}^{1}$ ; stationary length and energy therefore do not detect whether the space is CAT(1), they only detect the closed local geodesic.

Facts & Assumptions

Given: The circle $S_{ℓ}^{1} = R / ℓ Z$ of circumference $0 < ℓ < 2 π$ with its intrinsic metric; a fixed $n \geq 5$ ; the cyclic tuple $x$ of equally spaced points $j ℓ / n$ ; $l = ℓ / 4 - ε$ with $0 < ε < ℓ (1 / 4 - 1 / n)$ and $ℓ / n \leq h < l$ .

[L1]

Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences (vi): the circle $S_{ℓ}^{1}$ is compact and locally CAT(1), the distance between two points is the minimum of the two arc lengths, and $S_{ℓ}^{1}$ is not CAT(1) for $ℓ < 2 π$ .

[L2]

Uniform local radii, cyclic small-mesh polygons, mesh, length, energy, the midpoint operation and the zero-limit basin and Existence of the uniform radius, continuity of the midpoint operation, the energy drop, its equality case, and convergence of zero-limit polygons: the midpoint operation $f$ by arclength midpoints, the quantities $mesh, L, E$ , the basin $C_{h}^{0} (n)$ and the equality case.

[L3]

Polygon transfer, the basin as the shrinkable class, and the short-loop criterion (iii)-(iv): the minimum $m$ of the lengths of isometrically embedded circles, and the equivalence between $X$ being CAT(1) and $m \geq 2 π$ .

[L4]

Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, Open ball, closed ball and sphere in a metric space, Principal inverse sine and inverse cosine: the metric axioms, balls, and the elementary circle computations.

[L5]

The Axiom of Choice: AC enters only through the suppliers [L2] and [L3]; the explicit circle computation is choice-free.

Verification

technique · explicit arc-midpoint computation

1.1L2L4algebra

The data of the tuple (i). Consecutive points $j ℓ / n$ and $(j + 1) ℓ / n$ are at distance $ℓ / n$ (the shorter arc has length $ℓ / n < ℓ / 2$ , so it is the unique geodesic), so $mesh (x) = ℓ / n \leq h$ and $x \in P_{h} (n)$ , $L (x) = n \cdot ℓ / n = ℓ$ and $E (x) = n (ℓ / n)^{2} = ℓ^{2} / n$ .

2.1step 1.1L2L4algebra

The midpoint tuple is the shifted tuple (i). The arclength midpoint of the arc from $j ℓ / n$ to $(j + 1) ℓ / n$ is $(j + 1 / 2) ℓ / n$ , so $f (x)_{j} = (j + 1 / 2) ℓ / n$ ; this is the equally spaced tuple based at $ℓ / (2 n)$ , i.e. a rotation of $x$ by $ℓ / (2 n)$ . The rotation is an isometry of $S_{ℓ}^{1}$ , so $L (f^{k} x) = ℓ$ and $E (f^{k} x) = ℓ^{2} / n$ for every $k$ ; in particular the iterated lengths do not tend to $0$ and $x \notin C_{h}^{0} (n)$ .

3.1step 2.1L2algebra

Straightness and the equality case (ii). Every consecutive triple of $x$ lies on the unique minimizing arc of length $2 ℓ / n < ℓ / 2$ because $n \geq 5$ , and its middle point is the midpoint of that arc; hence each triple is straight, which is exactly the equality condition of L2: the tuple is the list of $n$ equally spaced points of the closed local geodesic $S_{ℓ}^{1}$ . Since the basin contains no nonconstant straight equilateral tuple by The uniform energy decrement on the basin, bounded iteration, and the closedness of the basin inside the short polygon space (iii), this is consistent with step 2.1.

3.2step 2.1L1L2L3algebra

The short circle and its minimum (iii). By [L1] the circle is compact locally CAT(1), but not CAT(1). Pairs at distance $< ℓ / 2$ have a unique shortest arc. If an isometrically embedded circle has length $u$ , its opposite points have two distinct minimizing arcs of length $u / 2$ , so $u / 2 \geq ℓ / 2$ . The whole circle realizes equality, proving $m = ℓ$ . By L3 it is nonshrinkable, consistent with the basin-only decrement and the constant positive iterated energy of step 2.1.

3.3step 2.1L1L3algebra

The boundary case (iv). For $ℓ = 2 π$ the same computations give $mesh (x) = 2 π / n$ , $L (x) = 2 π$ , $E (x) = 4 π^{2} / n$ and $f (x)$ the shifted equispaced tuple, so its length and energy are stationary while $S_{2 π}^{1}$ is CAT(1); stationary length and energy therefore do not distinguish the two cases.

4.1step 1.1step 2.1step 3.1step 3.2step 3.3L5∎

Conclusion. Clause (i) is steps 1.1 and 2.1, clause (ii) is step 3.1, clause (iii) is step 3.2 and clause (iv) is step 3.3; AC enters only through the suppliers [L2] and [L3] ([L5]).

Depends on

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Sources

Martin R. Bridson and André Haefliger, Metric Spaces of Non-Positive Curvature (Springer Grundlehren 319, 1999; author-hosted PDF) (standard reference, not scraped)
B. H. Bowditch, Notes on locally CAT(1) spaces (Aberdeen preprint, 27 scanned sheets) (standard reference, not scraped)