Viterbo’s Conjecture as a Worm Problem

The paper proves the duality between volume minimization over A(T,α) and maximizing the minimal T-length of non-translatable closed polygonal curves at fixed volume (Theorem 3.18), and it also establishes the threshold identity via the characterization that the largest α with K ∈ A(T,α) equals min_{q∈F^{cc}(K)} ℓ_T(q), with equality of the closed-curve and polygonal versions (Proposition 8.2 and Theorem 1.11). These match the candidate’s core claims. The paper’s path uses compactness/closure of coverings, density of polygonal curves, and monotonicity/scaling facts (e.g., Propositions 3.15, 3.11, 3.14), culminating in Theorem 3.18. The candidate reaches the same equivalences via an alternative construction: a “universal tiny probing polygon” to prove the hard direction of the threshold identity, followed by scaling arguments for the minimax equivalences. This proof is materially different but can be made rigorous with a small fix: replace the (unnecessary) appeal to the Lipschitz continuity of h_K by the simple estimate ⟨u0,u⟩ ≥ 1 − O(∥u−u0∥^2) on the sphere to conclude z+εu ∉ K. With that tweak, the model’s approach aligns with the paper’s results. Paper support: Minkowski worm problem setup and A(T,α) definition, the minimax equivalence (Theorem 3.18), the key implication K ∈ A(T,ℓ_T(q*)) for minimizers (Proposition 3.15), and the threshold characterization with closed curves along with polygonal equality (Proposition 8.2; Theorem 1.11) . Auxiliary density/continuity used in the paper also supports the model’s approximation step .