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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/15c/f/solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 15C / f / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

For points on the unit circle, their convex hull contains the origin exactly when they are not all contained in an open semicircle. Equivalently, the shortest closed arc containing them has length at least

π

. Applying this to

V = U_{2}^{†} U_{1}

and using

N (V) = conv (spec V)

(1)

gives

0 \in N (U_{2}^{†} U_{1}) ⟺ θ (U_{2}^{†} U_{1}) \geq π .

(2)

Part (b) now proves the claimed perfect-discrimination criterion.

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