LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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The four point condition implies slim triangles

Statement

In a geodesic space satisfying the four-point condition with constant $κ \geq 0$ , every geodesic triangle is $4 κ$ -slim.

Facts & Assumptions

Given: Such a space, specified sides of a triangle $(a, b, c)$ , and $p \in [a, b]$ .

[F1]

The four-point hypothesis gives the product inequality with the same constant at every basepoint, by The gromov product inequality implies the four point condition.

Proof

1.1

Write $ℓ = d (a, b)$ , $t = d (a, p)$ and $α = (b ∣ c)_{a}$ . Suppose first that $t \leq α$ . Since $α \leq d (a, c)$ there is $q \in [a, c]$ with $d (a, q) = t$ . Products along a radial geodesic give $(p ∣ b)_{a} = t$ and $(c ∣ q)_{a} = t$ . Applying the product inequality first through $b$ , then through $c$ , gives $(p ∣ c)_{a} \geq t - κ$ and $(p ∣ q)_{a} \geq t - 2 κ$ . Hence $d (p, q) = 2 t - 2 (p ∣ q)_{a} \leq 4 κ$ .

F1givenalgebra

2.1

If $t \geq α$ , use $b$ as basepoint. Indeed $(a ∣ c)_{b} = ℓ - α$ by expansion, and $d (b, p) = ℓ - t \leq ℓ - α$ . The argument of step 1.1 with $a$ and $b$ interchanged gives a point on $[b, c]$ at distance at most $4 κ$ from $p$ . At $t = α$ either construction works.

step 1.1algebra

3.1

Thus every point of $[a, b]$ is within $4 κ$ of the other two sides; relabeling vertices proves this for all sides and all specified triangles. Zero side lengths require only $t = 0$ , and when $κ = 0$ the produced point equals $p$ . There were only finitely many segment choices.

step 1.1step 2.1∎

Depends on

The gromov product inequality implies the four point condition

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Druţu–Kapovich Lemma 9.32; conservative two-inequality constant (standard reference, not scraped)