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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-1/2/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 1 / 2 / d / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Let

p, q

lie in the closed ball of center

z

and radius

R

. On their unique geodesic, the Hadamard squared-distance inequality gives

d (z, γ (t))^{2} \leq (1 - t) d (z, p)^{2} + t d (z, q)^{2} - t (1 - t) d (p, q)^{2} \leq R^{2} .

(1)

Thus every point of that geodesic lies in the ball. By uniqueness there are no other geodesics to consider, so every closed ball is geodesically convex.

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