Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 4 / b / Solution

Let a complete fan in a $d$-dimensional real vector space have ray generators $v_1,\dots ,v_r$. Every cone of the fan lies in the positive hull $\mathrm{cone}(v_1,\dots ,v_r)$. If $r<d$, these vectors do not span the space. If $r=d$, either they still fail to span, or they are linearly independent and their positive hull is a proper strictly convex cone. In either case their positive hull cannot be the whole vector space, contrary to completeness. Hence $r\ge d+1$.

The bound is attained: take rays throughand cones generated by every proper subset of these rays. This is the simplex fan, which is complete and has $d+1$ rays. Therefore the minimal number is $d+1=\dim N_{\mathbb{R}}+1$.

Solved by gpt-5.6-sol high.