feat: add Kakutani fixed-point theorem eval problem#287
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This PR adds the Kakutani fixed-point theorem (§33 of Knill's "Some Fundamental Theorems in Mathematics", the set-valued generalization of Brouwer) as a new eval problem: every upper-hemicontinuous correspondence F from a nonempty compact convex K ⊆ ℝᵈ to itself with nonempty convex closed values has a fixed point x ∈ F x. Mathlib has only the Riesz-Markov-Kakutani representation theorem (a different theorem); the fixed-point theorem is not in mathlib. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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This PR adds the Kakutani fixed-point theorem as a new lean-eval challenge problem — §33 of Oliver Knill's Some Fundamental Theorems in Mathematics, the set-valued generalization of Brouwer (and the engine behind Nash's 1951 equilibrium-existence proof).
Every upper-hemicontinuous correspondence
Ffrom a nonempty compact convexK ⊆ ℝᵈto itself, with nonempty convex closed values, has a fixed pointx ∈ F x.mathlib's
grep -ri kakutanireturns only the Riesz–Markov–Kakutani representation theorem (a different theorem). The fixed-point theorem itself is not in mathlib. One auxiliary definition is shipped in the Challenge:IsUpperHemicontinuous(closed-graph form).🤖 Prepared with Claude Code